Linear Discrete Parabolic Problems

LINEAR DISCRETE PARABOLIC PARABOLIC PROBLEMS PROBLEMS

NORTH-HOLLAND MATHEMATICS MATHEMATICS STUDIES STUDIES 203 203

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LINEAR DISCRETE PARABOLIC PARABOLIC PROBLEMS

NIKOLAI YU. BAKAEV BAKAEV Moscow, Russia

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Preface In this book we give a systematic systematic approach approach to to the the study study ofof linear linear discrete discrete parabolic problems for which which we we examine examine the the question question of of well-posedness, well-posedness, in particular, of stability. stability. The The developed developed techniques techniques can can also also be be applied applied in the analysis of convergence convergence of of the the corresponding corresponding discrete discrete solutions, solutions, asas demonstrated in Chapter 7. 7. We We remark remark that that the the use use of of the the concept concept ofofevoevolution equation in discrete time time and and of of some some related related concepts, concepts, which which will will be be explained in the sequel, is is of of great great importance importance for for our our investigation. investigation. This This approach appears natural natural in in view view of of the the fact fact that that our our subsequent subsequent consideraconsiderations are restricted to studying studying the the parabolic parabolic case case only. only. We Weexpect expect that that the the reader will have gotten the the impression impression that that the the developed developed discrete discrete theory theory by no means is a direct analogue analogue of of the the modern modern theory theory ofof parabolic parabolic differdifferential equations in Banach Banach space. space. ItIt will will therefore therefore be be seen seen that that the the below below approach possesses its own own distinctive distinctive features features for for which which one one can can find find no no analogue in the continuous continuous case. case. On On the the other other hand, hand, in in what what follows follows we we emphasize frequently the parallels parallels between between the the present present discrete discrete theory theory and and the classical continuous one. one. Our Our assertions assertions of of general general character character are are further further used for examining certain certain discretization-in-time discretization-in-time methods methods as as applied applied toto ababstract parabolic equations, equations, and and as as such such we we consider consider Runge-Kutta Runge-Kutta methods methods as well as other widespread widespread methods. methods. Note Note that that the the achieved achieved results results can can be used for the study of fully fully discrete discrete approximations approximations to to parabolic parabolic partial partial differential problems that are are constructed constructed on on the the basis basis of offinite finitedifference difference oror finite element discretization discretization in in space. space. However However such such an an investigation investigation would would to describe describe the the necessary necessary details, details, and and we we give give up up require much more room to idea at at least least in in the the present present book; book; perhaps perhaps itit will will be be the realization of this idea one. the subject for another one.

It is well known that any any modern modern discretization discretization method method used used inin the the nununon-stationary problems problems can can be be equivalently equivalently rewritten rewritten merical treatment of non-stationary step method method with with respect respect to to the the time time variable, variable, under under in the form of a single step restrictions. In In fact, fact, this this circumstance circumstance isis an an essential essential inincertain reasonable restrictions. subsequent considerations. considerations. Moreover, Moreover, itit allows allowsus usto toexamine examine gredient of our subsequent

vi

PREFACE PREFACE

discrete problems at the the first first stage stage of of analysis analysis without without any any connection connection with with the starting parabolic problem problem that that has has been been discretized. discretized. It is generally acknowledged acknowledged that that the the questions questions of of stability stability and and converconvergence are the most essential essential for for numerical numerical analysis analysis of of discretizations. discretizations. In In parparticular, if the method is is stable, stable, any any numerical numerical errors errors which which arise arise during during the the calculating process do not not grow grow in in the the sequel. sequel. The The analysis analysis of of convergence convergence is closely related to that that of of stability. stability. Moreover, Moreover, the the so-called so-called equivalence equivalence theorems (see, e.g., Richtmyer Richtmyer & & Morton Morton [135]) [135]) allow allow one one to to derive derive error error estimates for both stable stable and and consistent consistent discrete discrete methods, methods, assuming assuming that that the data are smooth enough. enough. In In dealing dealing with with the the parabolic parabolic case case we wehave have the the possibility of working with with non-smooth non-smooth data data as as well. well. The The question question ofof stabilstability/convergence is then then settled settled by by using using aa priori priori inequalities inequalities related related toto the the property of strong stability stability of of holomorphic holomorphic semigroups semigroups (cf. (cf. Larsson Larsson [99]). [99]). When working with concrete concrete classes classes of of discretization discretization methods, methods, the the questions questions of existence and uniqueness uniqueness of of aa solution solution in in the the necessary necessary sense sense also also need need toto be settled, and then we we have have to to discuss discuss well-posedness well-posedness instead instead ofof stability. stability. Much work has been devoted devoted to to the the verification verification of of stability stability and and by by now now many different tools for for its its analysis analysis are are known. known. We We can can mention, mention, among among others, the Fourier transform transform method method (see, (see, e.g., e.g., Lax Lax & & Wendroff Wendroff [101] [101] and and Richtmyer & Morton [135]); [135]); the the method method of of separation separation of of variables variables (see, (see, e.g., e.g., Douglas & Gunn [66]); the the energy energy method method (see, (see, e.g., e.g., Kreiss Kreiss [98]); [98]);the the method method based on estimating the the Green's Green's functions functions (Brenner (Brenner & & Thomee Thomee [51], [51],Schatz, Schatz, 145]); and and the the method method Thomee, and Wahlbin [141], and Serdyukova [143, 145]); based on the maximum maximum principle principle (see, (see, e.g., e.g., Samarskii Samarskii [139]). [139]). Recently, many authors authors have have successfully successfully dealt dealt with with discrete discrete initialinitialboundary value problems problems thinking thinking of of them them as as initial initial value value problems problems for for difference-operator equations equations in in Banach Banach (in (in particular, particular, in in Hilbert) Hilbert) spaces. spaces. As As a consequence, the stability stability theory theory of of linear linear discrete discrete non-stationary non-stationary problems problems in Hilbert spaces is, at present, present, well well advanced advanced and and fairly fairly effective effective inin applicaapplications connected with the the finite finite element element and and finite finite difference difference methods methods as aswell well as with other approaches approaches (see, (see, e.g., e.g., Bramble Bramble et et al. al. [49], [49], Karakashian Karakashian [90], [90], and Keeling [92], Oden k Reddy Reddy [121], Samarskii [139], Thomee [161, 162], and references therein). However However related related analysis analysis within within Banach Banach space space settings settings is much more complicated, complicated, which which results results from from complexity complexity of of the the geometry geometryofof Banach spaces as compared compared with with that that of of Hilbert Hilbert spaces. spaces. Most Most ofof this this work work has been devoted to the the case case of of linear linear differential differential problems problems with with aa constant constant operator (see, e.g., Bakaev Bakaev [10, [10, 16, 16, 17, 17, 18, 18, 20, 20, 28, 28, 30, 30, 31, 31, 32, 32, 37], 37], Brenner Brenner & Thomee [51, 52], Crouzeix Crouzeix [60], [60], Crouzeix Crouzeix et et al. al. [61], [61], Crouzeix, Crouzeix, Larsson, Larsson, and Thomee [62], Lubich Lubich & & Nevanlinna Nevanlinna [104], Nitsche & Wheeler [119], Palencia [127, 128], Schatz, Schatz, Thomee, Thomee, and and Wahlbin Wahlbin [141], and Thomee [162]).

PREFACE

vn

In applications such a situation situation corresponds corresponds to to partial partial differential differential initialinitialboundary value problems problems with with time-independent time-independent coefficients. coefficients. We We also also reremark that certain results results for for discretized discretized linear linear integro-differential integro-differential equations equations have appeared (see, e.g., e.g., Bakaev, Bakaev, Larsson, Larsson, and and Thomee Thomee [40, [40, 41], 41], Sloan Sloan && Thomee [147], and Zhang [168]). Less is known for the case case of of non-stationary non-stationary problems problems with with aa variable variable (that is, time-dependent) time-dependent) operator operator in in Banach Banach space. space. The The first first achievements achievements in this direction have dealt dealt with with situations situations with with one one spatial spatial dimension dimension (see, (see, e.g., Thomee and Wahlbin Wahlbin [163]) [163]) or or with with concrete concrete families families of of discrete discrete probproblems (see, e.g., Gudovich Gudovich hh Terteryan Terteryan [78] [78] and and Sobolevskii Sobolevskii [151, [151, 152]). 152]). For For results of general character, character, we we mention mention our our work work [10, [10, 16, 16, 25, 25,26, 26, 27, 27,28] 28] carcarried out in the course of of Banach Banach space space settings settings and and covering covering aalarge largevariety varietyofof applications. 1 Also, problems with a splitting splitting operator operator have have been been considered considered [15, 34, 35], as well as problems problems with with generalized generalized input input data data [20, [20, 34, 34, 35] 35] and and problems with a variable variable stepsize stepsize [44, [44, 30, 30, 36]. 36]. Such Such results results allow allow one one to to exexamine, in particular, wide wide classes classes of of parabolic parabolic semidiscrete semidiscrete and and fully fully discrete discrete problems with variable coefficients, coefficients, constructed constructed on on the the basis basis of of discretization discretization in time by single step or or multistep multistep methods methods and and of of finite finite difference difference or or finite finite element discretization in in space. space. This This research research has has become become possible possible due due toto certain resolvent estimates estimates that that have have been been established established for for elliptic elliptic differential, differential, finite difference, and finite finite element element operators operators (see (see [156, [156, 14, 14, 29, 29, 38, 38, 43, 43,39]). 39]).

The present book actually actually gives gives an an unfolded unfolded and and considerably considerably extended extended exposition of our results [13], [28], results collected collected in in the the papers papers [10] [10] [13], [15] [15] [28], and [30] [36], of which which some some are are quite quite old old and and some some more more or or less less recent. recent. Most of the above cited cited works works in in that that direction direction are are either either announcements announcements of results or have been been published published in in editions editions almost almost unattainable unattainable for for the the readers outside of Russia. Russia. Although in this book we we deal deal only only with with linear linear problems, problems, our our achieveachievements are significant for for studying studying numerical numerical methods methods for for nonlinear nonlinear parabolic parabolic equations, and we intend intend to to demonstrate demonstrate in in our our future future work work how how the the asserassertions in this volume can can be be applied applied to to the the analysis analysis of of nonlinear nonlinear problems problems as well. Recent results results devoted devoted to to examining examining nonlinear nonlinear discrete discrete parabolic parabolic problems in Hilbert and and Banach Banach space space settings settings can can be be found, found, inin particparticular, in Akrivis & Crouzeix Crouzeix [4], [4], Akrivis, Akrivis, Crouzeix, Crouzeix, and and Makridakis Makridakis [5], [5], Akrivis, Karakashian, and and Karakatsani Karakatsani [6], [6], Akrivis Akrivis & & Makridakis Makridakis [7], [7], Beyn Beyn & Garay [47], Gonzalez Gonzalez et et al. al. [72], [72], Ostermann Ostermann k.k. Thalhammer Thalhammer [125], and *Not so long ago Gonzalez Gonzalez & & Palencia Palencia [73, [73, 74], 74], using using different different techniques, techniques, showed showed certain stability assertions assertions that that are are similar similar to to some some of of those those established established within withinour ourearlier earlier work.

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Ostermann, Thalhammer, Thalhammer,and andKirlinger Kirlinger[126]. [126]. Ostermann, considerations start start bybyexamining examininggeneral generalquestions questions stability Our considerations of of stability within the the framework frameworkofofBanach Banachspaces spaces (see Part This investigation within (see Part I). I). This investigation is, inis, in the core coreof ofthe thebook. book.The Themain mainworking working concept in Part is that of evofact, the concept in Part I isIthat of evolution equation equation inindiscrete discretetime. time.On Onthe theone one hand, modern (even implicit lution hand, anyany modern (even implicit and/or multistep) multistep)discretization discretizationmethod method which is applied a non-stationary and/or which is applied to atonon-stationary problem necessarily necessarilyreduces reducestotothe theCauchy Cauchyproblem problem evolution equaproblem forfor an an evolution equain discrete discrete time time (under (underreasonable reasonablerestrictions). restrictions).OnOn other hand, tion in thethe other hand, rewriting the the starting startingdiscrete discreteproblem problemin in form evolution equation rewriting thethe form of of an an evolution equation in discrete discrete time time allows allowsone onetotoobserve observeitsits most important features a more most important features in ainmore transparent way. way. As Asititwill willbebeseen seenbelow, below, discrete theory presented transparent thethe discrete theory presented in in looks very veryoften oftenrather rathersimilar similartoto modern analysis of abstract Part II looks thethe modern analysis of abstract dif- differential equations equationsininBanach Banachspace space(see, (see, e.g., Amann or Lunardi [108]). ferential e.g., Amann [9] [9] or Lunardi [108]). the same same time time itit possesses possessesitsitsown owndistinctive distinctive features which At the features which havehave no no analogues in in continuous continuoustheory. theory.Apart Apartfrom fromeverything everything else, hope analogues else, we we hope thatthat our investigation investigation makes makesa avaluable valuablecontribution contribution theory of discrete to to thethe theory of discrete semigroups, whichisisatatthe themoment momentonly only a development stage semigroups, which at ata development stage (see,(see, e.g.,e.g., Lyubich [110], Nagy Nagy & & Zemanek Zemanek [114], [114], and Nevanlinna Nevanlinna [116, [116,117, 117,118]). 118]). Lyubich [110], main problem problem totobebeconsidered consideredis isposed posedin in Chapter 1, where The main Chapter 1, where we we introduce the the necessary necessaryworking workingconcepts concepts subsequent in Chapters introduce forfor ourour subsequent useuse in Chapters 2, 3, and and 4. 4. Furthermore, Chapter Chapter 22isisdevoted devotedtotothetheanalysis analysis stability of the Furthermore, of of stability of the Cauchy problem problemfor foran anevolution evolutionequation equation discrete time. These results Cauchy in in discrete time. These results are are most crucial crucial for for the thewhole wholesubsequent subsequentexposition. exposition.WeWe distinguish between most distinguish between autonomous and and non-autonomous non-autonomouscases casesforforwhich which essentially the autonomous we we essentially use use different techniques. techniques. InInthe theformer formercase casethethe main tools of analysis different main tools of analysis are are the the 2 operator calculus calculus methods methods while the the study study ofofnon-autonomous non-autonomousequations equations operator while based on on the the use useofofcertain certaindiscrete discreteversions versions Gronwall's lemma. is based of of Gronwall's lemma. TheThe corresponding stability stabilityestimates estimatesare are then valid finite time intervals corresponding then valid forfor finite time intervals only.only. The considerations considerations ininChapter Chapter3 3are areof ofparticular particular interest concern interest andand concern discrete problemswith witha asplitting splittingoperator; operator; ideas employed in the discrete problems thethe ideas employed in the pre-preceding chapter are are further further developed developedforforexamining examining stability of such ceding chapter thethe stability of such problems. significantthat thatour ouranalysis analysis based assumption problems. ItIt isissignificant is is notnot based on on thethe assumption that the the separate separatecomponents componentsofofthe thesplitting splitting necessarily commute; avoiding necessarily commute; avoiding this assumption assumptionisisessential essentialforformany manyimportant important applications. In this chapter applications. In this chapter we also also consider consider aamore moreinvolved involvedsplitting splittingconstruction construction which appears, which appears, for for 2 such methods methodsusually usuallydeals dealswith withintegration integration over suitable contours applicationof such The application over suitable contours complex plane. plane. Throughout Throughout the book book the orientation orientation of any any contour contour is taken taken in the complex what is accepted accepted in Appendix AppendixA.2. according according to what

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example, when handling second second order order discretizations. discretizations. In Chapter 4 we study from from the the viewpoint viewpoint of of stability stability more more general general equations, that is, discrete discrete evolution evolution equations equations with with aa memory memory term. term. The The techniques used here allow allow us us to to show show stability stability on on finite finite intervals intervals only. only. The general results of Part Part II can can further further be be applied applied to to the the study study ofofdisdiscretizations in time of abstract abstract linear linear parabolic parabolic problems problems inin Banach Banach space. space. This program of action is is realized realized in in Parts Parts IIII and and III. III. The The investigation investigation inin these parts is based on reducing reducing the the starting starting discrete discrete problem problem to tothe the Cauchy Cauchy problem for an evolution equation equation in in discrete discrete time. time. We Wetherefore therefore discuss discusshow how to perform such a reduction reduction and and how how to to verify verify the the meaning meaning ofof not not clear clear rerestrictions on the operator operator in in question question ifif we we proceed proceed from from aa discretization discretization in time of the original differential differential problem. problem. In In Part Part IIII we we concentrate concentrate on on the study of single step methods, methods, and and as as such such we we consider, consider, in in the the concrete, concrete, the Runge-Kutta methods. methods. However However the the below below techniques techniques can can be be extended extended in order to examine stability stability for for other other families families of of modern modern single single step step methmethods (see, e.g., Butcher [53], [53], Hairer, Hairer, N0rsett, N0rsett, and and Wanner Wanner [80], [80], Hairer Hairer && Wanner [81], or Stetter [155]). [155]). Next, Next, in in Part Part III, III, in in order order to to reveal reveal further further possibilities of our approach, approach, we we discuss discuss some some other other ways ways ofof discretization, discretization, more precisely the ^-method, ^-method, some some operator operator splitting splitting methods, methods, and and the the linlinear multistep methods. In In this this investigation investigation we we stay stay along along the the same samelines linesofof analysis as above; however however our our exposition exposition in in Part Part III III isis not not as as detailed detailed as as that carried out in Part II. II. In Chapter 5 we consider consider Runge-Kutta Runge-Kutta discretization discretization for for the the abstract abstract parabolic Cauchy problem. problem. We We introduce introduce some some working working concepts concepts and and hyhypotheses intended for subsequent subsequent use, use, among among others, others, certain certain conditions conditions ofof Holder-continuity; meanwhile meanwhile some some auxiliary auxiliary assertions assertions are are established established here here as well. Also, we give a suitable suitable classification classification of of the the Runge-Kutta Runge-Kutta methods. methods. However the most important important point point isis that that we we show, show, under under the the sectorialness sectorialness hypothesis and under one one of of the the Holder-continuity Holder-continuity type type conditions, conditions, that that the starting discrete problem problem can can be be equivalently equivalently represented represented as asthe the Cauchy Cauchy problem for an evolution equation equation in in discrete discrete time time and and is, is,therefore, therefore, uniquely uniquely solvable. Next, in Chapter 6, we derive derive stability stability estimates estimates of ofthe the discrete discrete solutions solutions for different classes of Runge-Kutta Runge-Kutta methods. methods. These These results results are are achieved achieved by by showing certain resolvent estimates estimates and and Holder-continuity Holder-continuity type type results results and and by subsequently using the the general general stability stability results results established established inin Part Part I.I. As concerns convergence estimates, estimates, these these are are presented presented in in Chapter Chapter 77 under various restrictions restrictions on on the the Runge-Kutta Runge-Kutta method method in in question, question, distindistinguishing between the autonomous autonomous and and non-autonomous non-autonomous cases. cases. Also, Also, we wedo do not pass over the phenomenon phenomenon of of order order reduction reduction which which may may be be observed observed for for

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approximations of equations equations with with an an unbounded unbounded operator operator and and which which shows shows itself by the fact that the the classical classical order order of of accuracy accuracy isis not not really really achieved. achieved. Completing Part II in Chapter Chapter 8, 8, we we study study variable variable stepsize stepsize Runge-Kutta Runge-Kutta discretizations and give stability stability estimates estimates for for such such discrete discrete problems problems under under some (rather general) restrictions restrictions on on the the families families of of time time partitions partitions used. used. In the next three Chapters Chapters 99 — —1111,, constituting constituting Part Part III, III, we we focus focus on on showing stability only and and carry carry out out the the investigation investigation under under further further restricrestrictions and simplifications. This This narrowing narrowing of of scope scope allows allows us us to to concentrate concentrate on the most essential aspects aspects of of the the considered considered methods. methods. Our analysis in Chapter 99 concerns concerns the the (9-method, (9-method, and and we we show show that that certain refined stability estimates estimates are are possible possible ifif the the operator operator inin question question isis under stronger restrictions restrictions than than just just the the sectorialness sectorialness hypothesis. hypothesis. In Chapter 10, under the the assumption assumption that that the the operator operator in in question question can can be decomposed into the sum sum of of two two generally generally non-commutative non-commutative operators, operators, we study certain methods methods with with aa splitting splitting operator. operator. In Chapter 11 we deal with with some some classes classes of of problems problems constructed constructed by by means of linear multistep multistep discretization. discretization. Finally, in Part IV which which consists consists of of only only one one chapter, chapter, we we discuss discuss the the stability of certain approximations approximations to to linear linear integro-differential integro-differential equations equationsinin Banach space by applying applying the the results results of of Chapter Chapter 4. 4. Our results below will mainly mainly be be expressed expressed in in terms terms ofof stability stability and and convergence estimates in the the norm norm of of an an abstract abstract Banach Banach space. space. Throughout Throughout the whole exposition there there occur occur constants constants in in assumed, assumed, intermediate, intermediate, and and final estimates, whose sizes sizes are are unessential unessential for for our our analysis. analysis. They They will will be be denoted by C and c when subject subject to to C C> > 00 and and cc >> 0,0, respectively. respectively. Both Boththe the C's and c's may implicitly implicitly depend depend on on other other parameters parameters which which are are fixed fixed inin traced. At At the the same same time time we wetake takecare care the context; 3 this dependence is never traced. of giving an explicit mention mention whenever whenever there there may may be be aa danger danger ofof confusion. confusion. As already mentioned above, above, our our considerations considerations will willbe becarried carried out outwithin within the framework of Banach Banach spaces. spaces. Throughout Throughout the the book, book, given given an an abstract abstract Banach space X, \\ || stands for for both both the the vector vector and and operator operator norm norm inin X, X, denote the the space space of of all all linear linear bounded bounded operators operators on on while B{X) is used to denote denotes the the dual dual space space for for X, X, \\\\ ||* ||* the the corresponding corresponding X. Furthermore, X* denotes norm in X*, and , ) the duality pairing pairing between between X* X*and and X. X. By By II we wedenote denote the identity operator both both in in XX and and in in X*. X*. In In what what follows follows we we also also deal deal 3

A parameter £ is thought of of as as fixed fixed without without explicit explicit mention mention if,if, for forexample, example,ititappears appears with the words 'there exists exists aa number number ££ ...' ...' or or 'for 'for some some££ ...'. ...'. The The same sameapplies appliestotosubsets subsets of the complex plane. On On the the contrary, contrary, ifif ££ appears appears with with the the words words 'for 'for all all ££...' ...' oror 'for 'for any £...', it is by no means means fixed. fixed. In In doubtful doubtful cases cases itit isis explicitly explicitly specified specified ifif££isisfixed fixedoror not.

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xixi

with some concrete function function Banach Banach spaces; spaces; the the corresponding corresponding notation notation for for their norms will be introduced introduced immediately immediately as as required. required. Furthermore, Furthermore, given given a linear (generally unbounded) unbounded) operator operator £, £, we we denote denote by by Dom£ Dom£ and and RRaann££ its domain and range, respectively, respectively, while while £* £* denotes denotes the the adjoint adjoint for for ££ (if (if defined). Next, if | | is a seminorm, seminorm, the the symbol symbol Dom Dom || || isis used used to to denote denote its its domain which is always always assumed assumed to to be be aa linear linear manifold. manifold. For For future future needs needs we also define some useful useful classes classes of of unbounded unbounded operators. operators. In In order order to to do do this, we first denote, given 0 is set arbitrarily. The last case is formally arrived at if we put T — oo,N oo,N = = oo. oo. We We also use the notation &k — {tn

tn — nk for n — 0 , . . . , N — 1}.

In this part of the book all working objects (operators, (operators, seminorms, seminorms, etc.), etc.), except the space X itself, generally depend on k;1 this dependence will not mentioned explicitly. At the same time, given k, the dependence on tn, if generally available, will always always be shown explicitly by adding n as a subscript functions of discrete to the corresponding object. 2 In particular, (X)-valued functions time tn are used. Given such a function Zn : 0,^ —+ X, we denote dZn = k~1(Zn+\ — Zn), £n € fife. (This notation will will be employed below for scalar discrete functions as well.) The main object of consideration in the present part of the book will be the discrete Cauchy problem problem 8Yn + %nYn = QnFn

(ortnenk,

Y 0 = VKy°,

(1.1)

for some given %l n,£ln,9{ G B(X). The vector y° G X and the discrete function Fn : &k —> X are considered as the input data of problem (1.1). 1

In principle, we could let the space X depend on k as well, but this is not needed for our subsequent aims. 2 This means that every object object that that depends depends on tn will have n as a subscript, separated by a semicolon from any additional subscripts.

4

CHAPTER CHAPTER 1. 1. PRELIMINARIES PRELIMINARIES

The discrete function YYn : £)& —> 36 is to be found. found. YYn is therefore thought of as a solution of (1.1). (1.1). In the case T < oo, we we employ employ the the condition condition 21 N — 21JV-I- This is not an important restriction, restriction, since since only only the the values values of of 2t 2tn, n = 0 , . . . , N — 1, are essential, but it will be convenient convenient in in our our subsequent subsequent statements. statements. Clearly, problem (1.1) can can be be equivalently equivalently rewritten rewritten as as follows follows

which is an explicit single single step step equation equation with with the the transition transition operator operator

il n = / - f c 0 n .

(1.3) (1.3)

It is therefore easily seen seen that that problem problem (1.1) (1.1) isis uniquely uniquely solvable solvable and and its its solution Y n is given by 3 n-l 0

Yn = Un-lflXy + kYsUn-hj+l&jFj,

(1.4)

3=0

where, for 0 < j , n < N, lj

if if

jn.

In spite of the fact that that the the unique unique solvability solvability of of problem problem (1.1) (1.1) isis evident, evident, itit is important to establish establish stability stability estimates estimates for for the the solution solution of of (1.1), (1.1), which which is the main objective in in this this part part of of the the book. book. As far as possible applications applications are are concerned, concerned, note note that that all all modern modern disdiscretization methods (including (including implicit implicit and/or and/or multistep multistep ones) ones) can, can, under under certain reasonable restrictions, restrictions, be be equivalently equivalently written written in in the the form form (1.1) (1.1) oror (1.2). In particular, for Runge-Kutta Runge-Kutta methods methods this this fact fact isis shown shown inin Section Section 5.3 while for linear multistep multistep methods methods this this observation observation isis pointed pointed out out inin Section 11.1. Now we give a series of of working working definitions. definitions. Definition 1.1. A difference difference equation equation of of the theform form (1.1) (1.1) isis called calledan anevolution evolution equation in discrete time. time. The The operator operator iinj iinj given given by by (1.5) (1.5) isis called called the the discrete evolution operator operator for for problem problem (1.1) (1.1) or, or, equivalently, equivalently, for for problem problem (1.2). In the particular case when when 2t 2tn = 21 and On = il do not not depend depend on on ttn, semigroup generated generated by by the the operator operator21. 21. iinfl = il n is called the discrete semigroup Throughout the book we we use use the the convention convention 52?=i 52?=i

== 00 ifif nn (r) = T>(0; r) for z G C and r > 0. Definition 1.2. A set TT C CC C is is called called aa set set of of Aj-configuration Aj-configuration with with some some integer J>lifTis

the the union union of of aa disk disk V(l\ V(l\ a) a) and and of of JJ sets sets ofof the the form form

a3 = {A : | arg(A - Q - a r g ( l - 0 ) 1 < P} D 0 ( 0 ; d),j = l,...,

J,

s o m e a € ( 0 , 1 ) a n d /3 G (0, a r c s i n a ) , where d = cos/3 — ( a 2 — sin 2 /3) 1 / / 2 and ^i = 0, ^2, , O are different points lying on the circle circle dV(l; dV(l; 1). 1). The The constants a and (5 are called called the the parameters parameters of of the the set set TT and and the the points points Q, Q, j — 1,. .., J, the singular singular points points of of T. T. For instance, when J — — 1, 1, we we deal deal with with aa set set of of Ai-configuration Ai-configuration which which has only one singular point: point: £1 £1 == 0. 0. Such Such sets sets will will be be of of particular particular interest interest in our subsequent considerations. considerations. Figure Figure 1.1 1.1 gives gives an an example example ofof aa set set ofof A3-configuration. Definition 1.3. An operator operator ££ n G B(X) is said to satisfy satisfy the the Aj-condition Aj-condition with some integer J > 11 ifif there there exist exist numbers numbers KK >> 0, 0, 88 >> 0,0, and and aa set set TT of A j-configuration such such that that (i) for all k G (0, K], the the operator operator k£ k£n has its spectrum contained contained in in the the set set IntT f c U{0} ; where fc / 5 / c ) u T , (1.6) (1.6)

6

CHAPTER 1. PRELIMINARIES CHAPTER 1. PRELIMINARIES

and and satisfiesthe theresolvent resolvent estimate, that is, for G (0,K], Gtflk, flk, (ii) ££n satisfies estimate, that is, for all kallGk(0,K], tn G (ii) and \glntrk, and \glntr jA — O ll~~ ,,

(1-7) (1-7)

3=1 3=1

where Q, jj — —1 , ......,, J, are arethe thesingular singular points points of of T. T. //// (1.7) (1.7)holds holds for for all all where Q, the operator operator £ n is is said said to to satisfy satisfy the the Aj-condition. Aj-condition. Any Any set set TT X fi I nnttTT,,4 the of A j-configuration, j-configuration, which whichmay may be beemployed employed in in the the above above context, context, is is called called Aj-proper fAj-proper,) fAj-proper,)for for the the operator operator £„. £„. IfIf TT isisAkj-proper j-proper (Aj-proper) (Aj-proper) Aj-proper the singular singular points points of of T T are arealso also called calledthe the singular singular points points of of the the for £ n ; the operator operator £ n . Re m maarrkk 1.1. 1.1. ItIt follows follows by by duality duality that that if an an operator operator £ n satisfies satisfies the theA Ajj-condition (the AAjj-condition) -condition) in in X, X, the thesame same is is true true for for its its adjoint adjoint £^ £^ in in condition (the X*. X*. The above aboveconcepts conceptsare areof of great importance for our subsequent considThe great importance for our subsequent considin2t(1.1) (1.1) erations. One Oneofofthe themain mainrestrictions restrictions below on the operator erations. below on the operator 2tn in will be be that that 2t2t satisfies the theAjAj-ororAj-condition Aj-condition with some integer will with some integer J > J1, > 1, n satisfies and the the situation situationwhen whenJ J—— \ will of particular interest. Atsame the same and \ will be be of particular interest. At the time some some seminorms, seminorms,whose whose properties related to above the above Aj- and time properties are are related to the Aj- and Ajconditions,are areused. used. Aj- conditions, Definition 1.4. Let Letthere there be begiven given an an ££n G B(X) B(X) and andfixed fixed numbers numbersJ G G N, N, Definition 1.4. m G GN, N, ££>> 0,0,with with m > > max{l,£}, max{l,£}, and and let let |||||| ||||n and and |||||| |*|*; nn be be seminorms seminorms andX*, X*,respectively. respectively. The Thepairs pairs (||| (||| ||| n ;£n) ;£n) and and (||| (||| ||*; ||*;n ;£n) ;£n) are arecalled called on X and and Aj(*;^|m)-concordant, Aj(*;^|m)-concordant, respectively, respectively, if there there exist exist Aj(£|m)-concordant if Aj(£|m)-concordant and numbers K > > 00 and and55>>0 0and anda a ofj A -configuration j suchthat thatthe the numbers K setset T T of A -configuration such fe spectrum of the the operator operator k£, k£,n is contained contained in in the the set setInt IntTT {0} whenever whenever spectrum of U {0} (0,K], where whereTk is is defined defined by by (1-6), (1-6), and and the therespective respective inequalities inequalitieshold, hold, k G (0,K], fc for all all kk G G(0,K], (0,K], tn G GU Uk, X X ££ IntT IntTfc , uG GX, X, and andxxGGX*, X*, j

|(AJ k£n)-mu\in < > 0, 0, with with m m> > max{l,£}. In addition, let ||| || n and I |*;n be two seminorms defined defined on on XX and and X*, X*, respectively. respectively. The The triplets triplets k (III ' Iln;£n;23n) and (1 |*;n;£n;2Jn) |*;n;£n;2Jn) are are called called A j(£\m)- and A^(*;£; \m)concordant, respectively, respectively, ifif there there exist exist numbers numbers KK >> 00 and and 55 >> 00 and and aaset set TT of A j-configuration such such that that the the spectrum spectrum of of the the operator operator k£ k£n is contained in the set lntTk U {0} whenever k £ (0, (0, K], K], where where TTfc is defined by (1.6), and the respective inequalities inequalities hold, for all k G (0,-JsT], t n G fifc, A ^ IntT f e ; u G X, andx& X*, J

k£ n)-mxnu\u < ck-t J2I A - 0 l ^ m \H

(i-io)

\i(\*i - fc£;rmQ?;x«l*;n < cfc-« J2IA - 0 l c " m llxll*.

(i-ii)

I( and

where Q, j = 1 , . . . , J, are are the the singular singular points points of of T. T. IfIf (1.10) (1.10) and and (1.11) (1.11) hold for all A £ I n t T , tfie tfie tripiets tripiets (|(| !„;£„; !„;£„; 9J 9Jn) and (||| !*;„;£„; 2?n) are called Aj(£\m)- and Aj(*\^\m)-concordant, Aj(*\^\m)-concordant, respectively. respectively. Any Any set set TT ofof AjAjconfiguration, which may may be be employed employed in in the the above abovecontext, context, isis called calledAj(.. Aj(.. .).)proper fAj(.. .)-properj for for the the corresponding corresponding triplet. triplet. IfIf the the size size ofof mm isis not not important in the context, context, m m does does not not appear appear in in the the above above notation. notation. Furthermore, we give a collection collection of of some some assertions assertions involving involving the the aboveaboveintroduced concepts and and intended intended for for subsequent subsequent reference. reference. 6

For example, we then write write simply simply Aj(*;£) Aj(*;£) instead instead of of Aj(*;£,\m). Aj(*;£,\m).

8

CHAPTER CHAPTER 1. 1.

PRELIMINARIES PRELIMINARIES

Lemma 1.1. If an operator operator ££ n G B(X) satisfies the Aj-condition, Aj-condition, itit then then follows that ||fc£ n|| < C for all k G (0, K] K] and and ttn € fifc. Proof. This fact can easily be shown shown by by using using the the operator operator calculus calculus formula formula (A.2) with /(A) = A and and with with k£ k£n substituted for £, where where the the contour contour H is taken independent independent of of kk and and bounded bounded away away from from the the disk disk T>(1; 1). We therefore have

f \(\I = J_ 2m J E

and a trivial estimation estimation shows shows the the claim. claim. (0,K] Lemma 1.2. Suppose that that ££ n € B(X) while ||fc£ n|| < C for all k G (0,K] andtn G fife. Lei a/so seminorms seminorms ||-| ||-| n and |-|||*;n /OTTTI respectively Alj(iy\m)and Aj(*;7*|m)-concordant Aj(*;7*|m)-concordant pairs pairs w£/i w£/i £„ £„ /or /or some some J,J, m m GG NN and and 7,7* 7,7*GG [0, m]. Then, with any fixed fixed dd G G [0,7] [0,7] andd* G [0,7*], the seminorms k"9 | - | n and fc''*! |*. n m/Z /orm respectively Aj(7 Aj(7 — i9|m)- and Aj(*;7* — i?*|m)concordant pairs with ££ n . Furthermore, for all kk G G (0, (0, if] if] and and ttn G fifc, F sup ffluln + A:7* sup |xi*;n < C.

(1.12) (1.12)

Proof. We will show the claim claim in in the the case case of of the the seminorm seminorm k^\l k^\l |||| n only, noting that the needed result result for for fc^ fc^1 |* ;n can be established in a similar similar manner. Accordingly, we we will will prove prove only only that that the the first first term term on on the the left-hand left-hand side of (1.12) can be estimated estimated as as stated. stated. Let R > 0 be denned by by R=2

sup

ke(o,K],tnenk

||A;£n||. ||A;£n||.

Using a standard argument, argument, itit follows follows that that for for |A| |A| >> R, R, since since lAI^Ufc^H lAI^Ufc^H O> s o * n a t with some 5 > 0 and with T fe given by (1.6), for all A £ Int TTk and ueX, j

|A - or m IMI-

(i.i4)

1.1. PRELIMINARIES

99

The last inequality yields, yields, under under the the additional additional restriction restriction |A| |A| 0, for all k G {0,K}, tn E Clk, A ^ Int (£,/, UV(5)), uu G G X, X,

and

Then, for any fixed d G [0,7] [0,7] and and ** G G [0,7*], /or aH k G (0, -ff], i n

and

10

CHAPTER 1. PRELIMINARIES CHAPTER 1. PRELIMINARIES

The proofis is similar similar to to that that of of Lemma Lemma1.2. 1.2. The proof Lemma 1.4. Let Letan anoperator operator£„ £„ satisfy satisfythe theA^-condition A^-conditionwith with some >1. 1. Lemma 1.4. some J> T/ien i/iere ezisi ezisia a ip ipG G(0,7r/2) (0,7r/2) and and aa/ii /ii>>0 0suc/i suc/i i/iai i/iai/or /or aZ ZZZk G G(0, (0,if], if], T/ien i/iere Gf2fc, M M> > Mi, Mi,a™d a™dAA Int(E^,U tn G g gInt(E^,

|A| ||(XI -- Z^W Z^W ++ !!(£„ +J) (A/ (A/-- ££n )- 1 || < < C. C. |A| ||(XI !!(£„ +M M 7/£ satisfiesthe theA j -condition, -condition,then then the thelast last estimate estimateholds holds with [i\ = =0. 0. 7/£ n satisfies with [i\ Proof. isobvious obviousthat thatthe thefirst first term termon onthe theleft-hand left-handside sideof of the theinequality inequality Proof. It is can be estimated estimatedas as needed, needed,with withsome some G (0, TT/2) and and[i\ [i\>>0.0.Using Usingthis this ip ip G (0, TT/2) can be the identity identity fact together togetherwith withthe fact x + fil) fil)(XI (XI- -££ +(A (A+ /x) /x)(XI (XI- - Sin)Sin)-1 (£ n + n ) - = -- II +

leads to the thedesired desired estimate estimatefor for the thesecond second term termas aswell. well. leads to thecase case where where£ n satisfies satisfiesthe theAj-condition, Aj-condition,it it is isnot nothard hard to to see seethat that In the for ji\ ji\==0.0. the sameargument argumentalso also works the same works for Lemma 1.5. Let Letan anoperator operator£ n satisfy satisfythe thekkj-condition j-conditionwith withsome some integer Lemma 1.5. integer J> > 1. 1. Then Then there thereexists existsa pb\ pb\ > > 00 such such that thatwith withany anyfixed fixed/J,Q /J,Q G GR, R,/or /orall all fj, > > Mi, Mi,& G (0, (0,iiff]], ,t n G Gfifc; fifc; u u ££ X, X, and andx xGG3£*, 3£*,

||«||0, 0, with some some/xi with 1

00 such such that that with with any any fixed fixed /xo /xo GGM, M, for all k G (0, K], t n G fifc, and A £ Int (T U V(5k)), V(5k)),

Proof. By the given conditions, conditions, the the operator operator ££ n satisfies (1.7) for k G (0, (0, K] K] and A ^ Int T, |A| > 6k, 6k, with with JJ == 11 and and some some 66 >> 0.0. With With this this fact fact inin mind mind and using the identity + /i 0 /) (A/ - fc^)-1 = - 7 + (A + /iOfc) (AI - fc-C)- 1, as well as taking into account account that that |A| |A| >> 6k 6k we we get get the the stated stated result. result.

Lemma 1.8. Suppose that that seminorms seminorms |||||| ||||||n and |]j |* ;n form respectively Akj(£\m)- and k kj{*;£\m)-concordant pairs pairs with with an an operator operator ££ n G B(JE) for some J G N, m G N, and and ^^ G G [0, [0,m]. m]. T/ien T/ien7 wii/i any fixed £i G [^,?n], [^,?n], the pairs (||| | n ; £ n ) and (||| l* ;n ;£n) wi^ be Aj(^i|m)- and miconcordant, respectively. respectively. Proof It suffices to note note that that one one can can take take the the 66 in in Definition Definition 1.4 1.4 to to be be strictly positive. Observe that if an operator operator ££ n fulfils the Aj-condition, Aj-condition, itit follows follows from from Lemma 1.4 that with some some ip ip G G (0,TT/2) and with any fixed IJ,O > 0 sufficiently large, for all k G G (0, (0, K], K], ttn G ftfc, and A

Therefore, if ^o > 0 is is sufficiently sufficiently large, large, the the operator operator ££ n + [IQI is sectorial, it has a bounded bounded inverse, inverse, and and its its fractional fractional powers powers (£ (£ n + —oo < £ < oo, are defined defined and and bounded bounded on on ££ (see (see Appendix Appendix A.3). A.3).

12

CHAPTER CHAPTER 1. 1.

PRELIMINARIES PRELIMINARIES

Lemma 1.9. Let an operator operator ££ n satisfy the Aj-condition Aj-condition and and let let seminorms seminorms and A^(*;7*|l)-concordant A^(*;7*|l)-concordant pairs pairs I || n and I |* ;n form respectively Aj(7|l)- and with £ n , for some J G N and 7,7* 7,7* G G [0,1). [0,1). Then Then there there exists exists aa/xi /xi >> 00 such such that with any fixed 71 >> 77 and and 7*1 7*1 >> 7*, 7*, for for all all kk GG (0, (0,K], K], ttn G fifc; u G X, and x G 3-*; (1.19) «|| (1.19)

Proof. We will show only (1.19), (1.19), noting noting that that the the second second estimate estimate can can be be proved in a similar way. way. Also, Also, by by the the convexity convexity inequality inequality (A. (A.18) 18) itit follows follows that ||(£ n + H\I)~^\\ < C for any fixed [i\ > > 00 sufficiently sufficiently large large and and for for any any fixed £ > 0. Therefore Therefore without without loss loss of of generality generality itit can can be be assumed assumed that that 7i < 1Let i\) G (0, TT/2) and 6 > 0 be the same as as in in the the assertion assertion of of Lemma Lemma 1.6 1.6 stated for £ = 7, and let let [i\ [i\ > > 00 be be aa fixed fixed number number such such that that |A |A — pL\\ > 5 for A c(|A| + 1) for A G F, taking next the seminorm seminorm |||||| |||| n of both sides and using Lemma Lemma 1.6 1.6 yields + ^iiyilu\L(5)), uGX, uGX, and and xx €€ X*, X*,

KAI-CCn + Srtn))-1?*!,, ^ C I A ^ I H I ,

(1.21) (1.21)

1.1. PRELIMINARIES

13 13

ffl(AJ - (£*n + m*n)rlX\Un

< ClAr-^Hxll*,

(1-22) (1-22)

and

IKAI-OCn + J O T O r ^ C CllA Arr 1 .

(1.23)

Proof. We will show (1.21) only, noting noting that that the other estimates (1.22) and (1.23) can be proved in a similar manner. Let 7' and 7* be fixed numbers such that that 7' G (7,1], 7I G ( T * , 1 ] , and 27' + 7* — 7 < 1 + d. By (1.20) and Lemma 1.9, there exists a Mi > 0 such that for all t n G fife, u G X, and x G X*, ||

(1.24) (1.24)

and

|||xllU;n 0 sufficiently large we obtain '-1.

(1.29)

14

CHAPTER PRELIMINARIES CHAPTER 1. 1.PRELIMINARIES

Applying then the above above argument argument as well as (1.20), (1.20), (1.24), (1.24), (1.25), (1.25),(1-27), (1-27), Applying then (1.29), it follows follows that that with with the same same #' as as above, above, for all all A A^^ Int Int(E^, (E^,UU and (1.29), l V(S)) and and uu ee X, X, since since 27' + + 7^ 7^-- 77 >8~8~ , sup 1 (XI -- £ n )- 1 9Jt n (A/ - ( ££n + IMI=i IMI=i

miy'(\I - (£ (£n ||(£n + miy'(\I With this result result in mind, mind, the claim claim now now follows follows from from (1.28) (1.28) with with both both With this right by u, u, using using a simple simple estimation estimation and and taking taking multiplied from from the right sides multiplied account Lemma Lemma 1.6 and and the thefact that that the pair (||| || n ;£n) ;£n) is Aj(7|l)Aj(7|l)into account concordant. concordant. D D future needs needs we examine examine here here also also the behaviour behaviour of fractional fractional For our future operator in question question is under under perturbation. perturbation. powers when the operator powers when Le m m maa 1.11. 1.11. Suppose Suppose that that seminorms seminorms ||| ||||n and and |||||| |* |*;ri form respectively respectively ;ri form Aj(7|l)andA\{*\^*\l)-concordant A\{*\^*\l)-concordant pairs pairswith with£ n G B(X), B(X), ££n itself itself is under under Aj(7|l)- and )-condition, and and DJln € B(X) B(X) satisfies satisfies condition condition (1.20), (1.20), with withsome some the A A1)-condition, N, i9 i9 G G [0,1], [0,1], and and 7,7* 7,7* G G [0,1) [0,1)such such that that 7 + 7* 7* > 00 sufficiently sufficiently large largeand any any fixed fixed £ G [0,1 [0,1++ ## — 7*), 7*), with any £, € [0,1 [0,1++ 00 -- 7), 7),for forall allkkGG(0, (0,K), K),tn t G Q Qk, u G GX, X, and andxx GG 36*, 36*, || (£n + + Win + Hil)*u\\ Hil)*u\\ < C C II (£„ + + W J)*u|| J)*u||

(1.30) (1.30)

and Proof, demonstrate how how to to prove prove (1.30) (1.30) since since the second second estimate estimatecan Proof, We demonstrate derived in aa similar similar manner. manner. Also, Also,note notethat thatthe case £ = 00 is is trivial, trivial, so be derived it can be be assumed assumed that that £ > 0. 0. Let if) if) and SS be be chosen chosen as in in Lemma Lemma 1.10 1.107 and let let 7' 7' be beaa fixed fixed number number G (7,1) (7,1)and and7'7'++7* 7* 00 sufficiently sufficiently large, large, for all all A A^^ Int Int(E^, (E^,UUT>(8)) T>(8))and uu G GX, X, any fixed

< C||(£n+ C||(£n+ M1-0 M1-07 '(A/-£nrM '(A/-£nrM 7 + UlrfuW UlrfuW < C||(£ C||(£n + /Xl/) '^(A/ - fin)"1!! ||(£n + 1 Kf M M (1-31) (1-31) 7 assume without without loss lossof generality generality that that tp and and 55 are areare arechosen chosen as in in Lemmas Lemmas We can assume 1.3 and and 1.4 1.4which which will will be applied applied together together with withLemma Lemma1.10 in in the thepresent present proof. proof.

1.1. PRELIMINARIES 1.1. PRELIMINARIES

15

At thesame sametime, time, if££G G(7,1 (7,1+ + — —7*), 7*),since since £nn satisfies satisfiesthe theAj-condition, Aj-condition, At the if £ usingagain again Lemma 1.9yields, yields,for forall all A^ ^ Int Int(E^, (E^,UUV(5)) V(5)) anduue e 1.9 A and X,X, using Lemma Zn^uln < < \I(\I- -Zn^uln \I(\I ^^

zii/^uH. ++zii/^uH.

(1-32)(1-32)

byLemma Lemma1.1 1.1we wehave have Next note that Next note that by \\k£n\\55for for A^ ^ Int IntE^. E^.We We then apply theidentity identity A— IA A then apply the 11 (£nn + +m mnn + +M MII/ /) ) ")11" -- ((A A/ -/ (£n (£n+ + Mi-Or Mi-Or {xi -- (£ {xi /))-11OT OTnn(A7 (A7-- (£ = (A/ (A/-- (£ (£nn+ +m mnn + +/xx11/))(£nn+ +M MII/ /) ) ")11", = 88 and the theestimates estimates(1.20), (1.20), (1.31), and(1.34) (1.34) toobtain, obtain,for forall all A£ £ and (1.31), (1.32),(1.32), and to A IntE,/, IntE,/,and anduu€ € X,X, since sincekk < >c(|A| c(|A|+ + 1), 1), 11 11 I -- (£ (£nn + +m mnn + +Mi/))" Mi/))" -- (AI (AI-- (£ (£nn+ + Mi/))" Mi/))" ) «|| «||

1 ? 22

Cfc^'|A|«(|A| l ) > l' )' >| '||((££ Cfc^'|A|«(|A| s

ifif ^^G [ oo, ,7 ] ,,

(7,1+ +^^-7*)-7*)/ /HH l lif if £G £G (7,1 (1.35) (1.35)

weuse use the the last last three threeestimates estimates with with A— — /n /n A in inplace placeof ofA, A,for forA A^ ^ Int Int£^. £^. In fact, fact,we In

15

16

CHAPTER CHAPTER 1. 1. PRELIMINARIES PRELIMINARIES

Next we denote R = 2sup fce(0tK ],t n en k (\\k(£>n+ViI)\\ + \\k9Jln\\). By Lemma 1.4, (1-23), and a standard argument, it follows that for all A ^ Int (S^, (~l | | ( A / - ( £ n + 2Jl n + M!/))- 1 1|+ | | ( A / - ( £ n + /i 1 /))- 1 1| 0 sufficiently large, the assertions of Lemmas 1.10 and 1.11 are still valid if one requires that 7,7* belong to [0,1] instead of [0,1).

1.1. PRELIMINARIES

17

In the remainder of the the section section we we want want to to show show that that under under reasonable reasonable conditions on the operator operator in in question question there there do do exist exist seminorms seminorms with with the the properties stated in Definitions Definitions 1.4 1.4 and and 1.5. 1.5. L e m m a 1.12. Let £ n satisfy the Aj-condition Aj-condition (Aj-condition) (Aj-condition) with with some some integer J > 1 and let Q, Q, jj = = 11,,......,, J, J, be be the the singular singular points points ofof ££ n . Then, (with fio fio == 00 in in the the case case of of the the AjAjwith any fixed JIQ > 0 sufficiently large (with condition), with any fixed m £ N, N, rj rj 66 N NU U {0}, {0}, and and ££ ££ [0,min{m,?7}] [0,min{m,?7}]; with J

(1.39) and with the seminorms |||||| |||| n and § |* ;n given by

H\n = and

the pairs (||| | n ; £ n ) and concordant (Aj(£\m)- and and

^ ) will be A^(£|m)- and and )-concordant), respectively. respectively.

Proof. For the sake of brevity brevity we we denote denote 5J 5J J;n = / — £ 1k£n, j = 2,. .., . . , J, so that we have QJn = TL-oOJ 7?,,. Our argument is based based on on using using certain certain smoothing smoothing properties properties ofof the the operators 2X,;n, j = 2 , . . . , J, and 5J n . Note first that the operators operators QJj;n are uniformly bounded ||QJj;n|| > 00 sufficiently sufficiently small and some 6 > 00 (8 (8 == 00 in in the the case case of of the the A ./-condition), for all j = 2 , . . . , J and A £ Int T, |A |A -- Cj| Cj| >> 5fc, 5fc, ||(A/-fc£ n )- 1 QJ j;Tl ||

6k, ||(£ n + fxolf (XI - fc£ n)-m5Jn||
1 and J G N if it satisfies the Aj-condition in which the constant C (see (1-7)) is replaced by M. Aj-proper sets for the operator £ are then defined in the same way as in Definition 1.3. 3

We then write simply 21 and and il instead of 2ln and il n , respectively.

26

CHAPTER 2. MAIN RESULTS ON STABILITY

In fact, employing the Aj(M)-condition means that our results may be used for situations in which the size of the constant in the corresponding resolvent estimate varies considerably. Now we state an analogue of Theorem 2.3 with £ = 0.4 Theorem 2.6. Let the operator 21 satisfy the Aj(M)-condition with some J G N and M > 1. Then, for all k G (0, K) and 1 = 0,1,..., ||il'|| 0 in the case J = 1 only. Theorem 2.7. Let the operator 21 satisfy the Ai(M)-condition with some Then, with any fixed £ > 0, for all k G (0, K) and I = 0 , 1 , . . . , M>\.

Proof. Unfortunately, we cannot act on the complete analogy of the proof of Lemma 1.12 which further implies Theorem 2.3 since in that proof we have used the convexity inequality which does not control the size of the stability constant. We will therefore use instead a slightly different argument which is helpful in proving directly Theorem 2.3 as well. As in Section 1.1, it follows from Lemma 1.4 that the fractional powers 2ln, £ > 0, are denned and bounded on X. Further, let a set T be Ai-proper for 21 and let F be the contour coinciding with dT. Using the operator calculus formula (A.17) with /(A) = (1 — A)', with A;2t in place of £, and with F in place of H^,5 we have, for I = 0 , 1 , . . . ,

Now it is not hard to see that the claim follows because of the reasonings already used in the proof of Theorem 2.1. Note also that Theorems 2.1, 2.2, 2.4, and 2.5 can be similarly developed. We leave the details for the reader. In the remainder of the section we intend to get stability estimates based on a slightly different resolvent condition. 4

In this section we present no estimates involving the adjoint operators. Clearly, the mentioned deformation of the original contour is tolerable by Cauchy's Theorem. 5

2.2. DISCRETE SEMIGROUPS: SEMIGROUPS: SOME SOME SPECIAL SPECIAL POINTS POINTS

27 27

Definition 2.2. We say say that that an an operator operator ££ G GB(X) B(X) satisfies satisfies the the STj(M)STj(M)condition® with some M M >> 11 and and JJ G GN N ifif its its spectrum spectrum isis contained contained inin the the shifted unit disk X>(1; 1) 1) whenever whenever kk G G (0, (0,K] K] and and

\\{\I - S,)- 1]] < M ^ l A - C j T 1

with Ci = 0, (l; IntX>(l; 1), 1),

, CJ given fixed fixed points points lying lying on on the the circle circle dT>{\\ 1).

On the one hand, if £ satisfies satisfies the the Aj(M)-condition, Aj(M)-condition, then then the the operator operator fc£ clearly has satisfied the the ST/(M)-condition ST/(M)-condition with with the the same same constant constant M. M. On the other hand, it can can be be shown shown by by analytic analytic continuation continuation (see (see Theorems Theorems A.I and A.2) that the STj(M)-condition STj(M)-condition for for the the operator operator fc£ fc£ has has implied implied the Aj(Mi)-condition for for the the operator operator ££ with with some some M\ M\ >> M. M. Therefore, Therefore, ifif we want to use the above above techniques techniques in in order order to to get get stability stability estimates, estimates, we we have to trace the dependence dependence M\ M\ on on M. M.

Theorem 2.8. Let the operator operator fc2t fc2tsatisfy satisfy the the STj(M)-condition STj(M)-condition J G N and M > 1. Then, Then, for for all all kk G G {0,K] {0,K] and and 11== 0,1,..., 0,1,...,

with with some some

||H'|| < C M l o g ( l + M).

Proof. For the sake of simplicity, simplicity, we we show show the the result result in in the the particular particular case case J = I. 7 Assuming therefore this this condition, condition, let let (p (p== arcsinl/(2M). arcsinl/(2M). Assume first that (I + + I)" I)"1 < 2sin 0, we we obtain in the case 0 < £ < 1, for 1 = 0 , 1 , . . . , \\h\\

< 4Af Jsiny) y

1

e x p ( - ( / + 1) simp (sirup - x/2)) dx Jsintp


l-M(2cos(argA))

1

M

1

Af- cos(argA)

cos(argA)' cos(argA)'

At the same time we have, have, for for A AG G FF^^,, |1 — A|* = (1 — (2 — 1/M)/M cos2 argA)' /2 < Cexp ( - ^ - t i cos 2 arg A).

Therefore, observing that l/M)cosi9exp(M?))/di?| 0, r«

if £ > i .

(( 22 2 5 )

-

Next, using the argument leading to (2.21) and (2.22), we get I ) - 1 if 0 < £ < l , 1)~^ if 1 < £ < 2,

( 2 2 66 ))

(2.27)

(2.26), and (2.27) that Clearly, in the case 0 < £ < 1 it follows from (2.25), (2.26), 2

^

ifMl _/+i sup| ^| ??11 and (2.45), Denotingthen then = tt n _/+i sup| M ||il n _ 11)) ^| and using using(2.43), (2.43), (2.45), n>l = M | =1 =1 ||il and Theorem 2.1,2.1, we (2.31), after after aa simple and Theorem we obtain obtainfrom from (2.31), simpleestimation, estimation, for fort;t;
d using using further Theorem 2.2 (2.61), we with the right by x> ax d further Theorem 2.2 andand (2.61), we get,get, with the right

ra-1 ra-1

q=l+l q=l+l

Since — 7* i9< < 1 71 and — $1 + 7* applying 1, 2.1, with — 1 Since 77+ + 7* — i9 1 and + 7* 71 — $1 6260k, that + 60I)\\
thus, for A g Int (T" U V(S ok)),

(3.6) is uniquely solvable and, It follows from (3.12) and (3.21) that equation (3.6)

for

\£lnt(r"UV(S ok)), < CIA!" 1.

(3.22) (3.22)

shows that that Furthermore, a simple calculation shows (XI - fc2l2;n) Go(A; z) (zl -

fc^)- 1

- (/ - m 2]n) G0(A; z) (zl - k^n)-1 fc2li;n = (zi - mx,n)-1 + (i - m2yn) GO(A; z). With this fact in mind and again using (3.7) (3.7) we we get, get, for A ^ Int (T"{JV(Sok)), (A/ - A:2l2;n) C-(I- fc2l2;n) C fc2li;n

= ^-. I {zl-m1.n)dz + ^-. I (I-m2,n)Go(\;z)dz

60

CHAPTER CHAPTER 3. 3. OPERATOR OPERATOR SPLITTING SPLITTING

PROBLEMS PROBLEMS

since the second integral integral vanishes. vanishes. ItIt isis then then not not hard hard to to see see that that by by the the definition of C and U we we have, have, for for A A££ Int Int (T" (T" U U V(S V(Sok)), {\I-k%n)1l = I.

(3.23) (3.23)

Similarly, it can be shown shown that that there there exists exists an an operator operator itit GGB(X) B(X) (depending (depending for A A ^^ Int Int (T" (T" U U T>(Sok)), T>(Sok)), on A) such that with 5Q > 0 sufficiently large, for n(\i-ksin) = i.

(3.24) (3.24)

Now we conclude_by (3.23) (3.23) and and (3.24) (3.24) (cf. (cf. Hille Hille & & Phillips Phillips [84, [84, Paragraph Paragraph 1.13]) that n = TZ= (Al-fcOn)(Al-fcOn)- 1 for A g Int (T"UX>(<J ofc)), which together with (3.22) shows that the the operator operator 2l 2ln satisfies the A^-condition A^-condition and and the the claim thus follows by Theorem Theorem 2.3. 2.3. Next we show a slightly slightly different different stability stability estimate estimate in in comparison comparison with with (3.4). This new estimate is of of independent independent interest interest and and isisused used in inthe the stability stability analysis of problem (1.1), (1.1), (3.1) (3.1) in in the the non-autonomous non-autonomous case. case. Theorem 3.2. Let the the conditions conditions of of Theorem Theorem 3.1 3.1 be be satisfied satisfied and and let let inin addition hypothesis C O M 1 [ K ] hold for some K G [0,1). Then, with any any fixed no > 0 sufficiently large large and and with with any any fixed fixed ££ GG [0,1], for all k G (0, K] and ti,tn G £}&, we have

3=1

Proof. Let /izo > 0 be chosen sufficiently sufficiently large large in in order order to to make make our our below below argument correct. In particular, particular, this this fact fact yields yields that that the the fractional fractional powers powers (2lj.n + iM)I)*, £ > 0, of the operators (2lj ;n + Mo-0, J = 1,2, are defined defined and and bounded on 36. Also, we we fix fix arbitrarily arbitrarily an an r\r\G G [0,1). [0,1). Let further C, T", F\, and and (?o(A; (?o(A;z) z) be be defined defined in in the the same same way way as as inin the the proof of Theorem 3.1. Using Using then then (3.8), (3.8), (3.9), (3.9), (3.17), (3.17), (3.18), (3.18), the the convexity convexity inequality (A. 18), and the the above above argument, argument, we weget, get, for for AA^^ Int Int (Y"u£>(/zo&)) (Y"u£>(/zo&)) > and z G T\,

||^(2li;n + toiyW - k^n)- 1 II < Clz^1

(3.26)

and

< C(|A| + H)"- 1 .

(3.27)

3.1. ESTIMATES

FOR DISCRETE SEMIGROUPS SEMIGROUPS

61

Also, by (3.27) and (3.7), acting as in the preceding proof and noting noting that that 7] < 1, we find, for A £ Int (T" U V(p, Qk)), ||^(2l2;n + fi0I)vC\\ < CIAI"- 1.

(3.28)

Similarly, changing the order of the resolvents on the right-hand side of (2.4) and further using the following identity, for £i, £2 €

Com (( £i)- 1 (AJ - £ 2 )- 1 Com (-C2I-C1) (A/ - £ 2 ) ~ 1 ( ^ - £ i ) ~ \

it is not hard to see, with the aid of (3.26), COMl[/t], (3.8), (3.8), (3.17), (3.17), and (3.20), that for A g Int (T" U V(fi ok)),

WW^n + ^iyCW^ClXr 1.

(3.29)

Therefore, by virtue of (3.28), (3.29), and (3.21) and using (3.6) we have, for A£lnt(T"uP(MoAO), A£lnt(T"uP(MoAO), P"(2l i ; n + Mo/) T '(A/-/c2l n )- 1 || £*> ||

(3.51)

and (3-52) Furthermore, with the aid of the following identity, where where we use the notation £ = Com (2li ;n |2l 2;n ), Com (3Li;n|3l2;n) = 4C - 2A;(2l1;n + 2l 2;n) £ - 2k€ (2l 1;n + 2l 2;n )

using (3.13), (3.51), (3.52), (3.52), and the above permutation techniques techniques we can prove that the hypotheses COMl[/t] and COM2[«;] are still satisfied with 2tj ;n substituted for 2lj ;n - We will show, for example, a possible estimation of the quantity |(x, A;2li;n 0. Also, since ||21 - k%. n\\ < C, we have for j = 1,2 and u e X, X,

It thus follows that the estimate (3.25) holds even even with with iin substituted for On, both for £ = 0 and £ — 1 and, and, applying (A.18), for any fixed £ € [0,1]. 14

Without loss of generality fj.o can be assumed to be positive and sufficiently large.

3.4. COMMENTS AND AND BIBLIOGRAPHICAL BIBLIOGRAPHICAL

REMARKS REMARKS

69 69

Therefore using the above above observations, observations, Theorem Theorem 3.5 3.5 now now straightforwardly straightforwardly follows. Also, employing employing analogues analogues of of the the identities identities (2.28) (2.28) and and (3.47) (3.47) inin the the case where iln plays the the role role of of On On and and following following the the proof proof of of Theorem Theorem 3.4 3.4 shows the assertion of Theorem Theorem 3.6 3.6 as as well. well.

3.4

Comments and bibliographical bibliographical remarks remarks

Sections 3.1 and 3.2: 3.2: The The present present considerations considerations follow follow in in essence essence our our earlier work [13, 15, 34]. 34]. Section 3.3: Similar results results for for the the unperturbed unperturbed problem problem (1.1), (1.1), (3.49) (3.49) have have been announced in [21]. [21].

This Page is Intentionally Left Blank

Chapter 4

Equations with Memory Memory In this chapter we extend the field of our study by including equations with a memory term in our scope. Such equations arise, arise, for example, from discretization of integro-differential equations equations (see Part IV). More precisely, instead of problem (1.1) we now deal with the following discrete Cauchy problem problem iYi + Fn

iovtnenk,

Y Q = y°,

(4.1)

1=0

with ty nj € B(3L) given for ti < t n (depending generally on k) and all other objects retaining their former former meaning meaning (see (see Chapter Chapter 1). In view of the many important applications (particularly, (particularly, in the case where 2l n depends on tn, but not exclusively) it is expedient to consider together with problem problem (4.1) its perturbed version, with with some some *B *Bn € B(X), lYl + Fn

iovtneClk,

Y 0 = y°. (4.2)

1=0

The problems (4.1) and (4.2) differ from (1.1) and (2.32), respectively, by the presence of the memory term SILo^Pn.*-^ SILo^Pn.*-^ m * n e corresponding equation. equation. 1 Obviously, both (4.1) and (4.2) are uniquely solvable. To the problems (4.1) and (4.2) we associate the problems for the corresponding homogeneous equations equations with with floating floating initial initial condition, condition, for tn,tj G fifc with tj < t n, ^n,iYi,

Misgiven,

(4.3)

1=3

the sake of simplicity throughout this this chapter chapter it is accepted that £!„ = 9\ = I.

71

72 72

CHAPTER 4. EQUATIONS EQUATIONS WITH MEMORY CHAPTER 4. WITH MEMORY

and and iYu Misgiven. Misgiven. iYu

(4.4) (4.4)

1=3 1=3

ofthese theseproblems problems notgive giverise riseto todoubts doubtsas as The unique unique solvability of doesdoes not The solvability well. Let 2Bnn,j,j and and2B 2Bnnj be bethe thesolution solutionoperators operators forthe theproblems problems (4.3) well. Let 2B for (4.3) and (4.4), respectively, is, that forYY thesolution solutionof of(4.3), (4.3), and (4.4), respectively, that is, for n the n Yn^VOn-ijYj, Yn^VOn-ijYj,

(4.5)(4.5)

and forYY thesolution solutionof of(4.4), (4.4), and for n the n Yn =Wn-uYj. Wn-uYj. Y n=

(4.6)(4.6)

Then, bylinearity linearityand andcausality causalityit itfollows followsthat that thesolution solutionY Ynn of ofproblem problem Then, by the (4.1)admits admits therepresentation representation the (4.1) n--ll n

Y = =2B 2B_i _iny° ny°+ +kkN NW W_i_i-+iF-+iFY

(4-7) (4-7)

3=0 3=0

thesolution solutiony ynn of of(4.2) (4.2)can can berepresented represented by whilethe be by while n-\ n-\

Y =W =W i ooy° y° i + +kk\ \ 22J 22J_i _i-+iF-+iFY

8) (4 (4 8)

It is iseasily easilyseen seen (4.7)and and (4.8) takethe thesame sameform form as(1.4) (1.4)and and (2.33), It thatthat (4.7) (4.8) take as (2.33), respectively. Therefore allwe we needfor forshowing showing stability isto tofind find aasuitable suitable all need stability is respectively. Therefore estimatefor forthe theoperators operators _ij and and2H 2Hnn_ij, _ij, which whichmeans means infact, fact, estimate 2U2U that,that, in nn_ij ouraims aimsit itsuffices sufficesto toanalyse analysethe theproblems problems (4.3)and and (4.4) only. for our (4.3) (4.4) only. for For furthersuccess success wenow nowderive derivesuitable suitable equations for2B 2Bnn_ij _ijand and For further we equations for 2U Since,by byanalogy analogywith with (1.4), forYY thesolution solutionof of(4.3) (4.3)we we have 2U nn_ij. _ij. Since, (1.4), for have n the n n n --l l

Y

I

I

—i tt nn_ _i i YY-++k ^S~^ ^S~^ k i li nl -ni - zi + iz y+ i^ y^ P^/ Y ^Y P / f fo or r t

(4-10) (4-10)

1=3 q=l q=l 1=3

where the discrete evolution for operator problem (2.32). where iinj iinj isis the discrete evolution operator for problem (2.32). Inthe the next two sections we willexamine examine the stability of the problems next two sections we will the stability of the problems In and (4.2) using the equations (4.9) and (4.10) and distinguishing be(4.1) (4.1) and (4.2) using the equations (4.9) and (4.10) and distinguishing between the cases where the operator 2l depends and does not depend on . tween the cases where the operator 2l and does not depend on ttn n. ndepends n The main idea employed isthat that the memory Y^i=j ^Pn,/^ isconsidered considered The main idea employed is the memory term term Y^i=j ^Pn,/^ is (nonlocal) perturbation to the principal l ^ or or 2(2l (2l t nn+ +^& ^& )Y asaa(nonlocal) perturbation to the principal term term 2t nnl^ as n)Y n.. n n Acting in such way leads to estimate discrete analogue Acting in such aaway leads to anan estimate whichwhich presents presents aadiscrete analogue of an integral inequality and solved be by by using suitable discrete verof an integral inequality and cancan be solved using aasuitable discrete version of Gronwall's Note that these are applicable techniques inthe the sion of Gronwall's lemma.lemma. Note that these techniques are applicable in case of finite intervals It is only. therefore assumed T >00there there exists existsT\ T\ G G(0,T] (0,T]such such that that whenever wheneverts-m < < T\, T\, ll vv^^nn- l-.l-.(-7( 37+37+»73»)3 ) a-73 a-73

4.1. THE GENERAL CASE

75 75

and n-ln-l

TTien the solution Yn of both (4-1) and (4-2) satisfies the stability estimate (2.81). Proof. In view of the representations (4.7) and (4.8) it suffices to show the estimate, for all u G X and tj < tn,

Since both terms on the left-hand side of (4.17) are similarly estimated, we will demonstrate our argument for showing the desired result for the first term only. To that end, we first note by Theorem 2.10 that with P — L(n + ?)/2j i forall u € X and tj 0 such such that that 73 73 l = (21 + / / o / ) " ^ , / , for all k E (0,K] and tj,t tj,tm,ts,tn

subject to tj < t m < ts < tn, for any 77 > 0 there exists exists T\ T\ G G (0,T] (0,T] such such that that whenever whenever tts-m

Zt=L (Wn-l,l\\ + k EU t-l t-lgWq,l - ^n-l,Hl) < V-

< T\,

(4-22)

We start by examining problem problem (4.1) (4.1) in in the the considered considered case. case. T h e o r e m 4.3. Let the operator^21 satisfy the K kj-condition with some J G GN N and let (4-22) hold. Then the solution Y n of problem (4-1) satisfies satisfies the the estimate, for all k G (0, (0, K] K] and and ttn G fifc; (4-23)

4.2. A PARTICULAR

CASE

77 77

Proof. By (4.7) it suffices to show that ||2Bn-ij|| < C for all tj < tn.

(4.24)

Applying now (4.9) with with 2l n = 21 gives, after a simple calculation, with with?($ ?($nj defined above, for tj < t n, n—ln—l

Wn-u

= an~j + k J2 ^ it"-"-1 (21 +mi) (yq>l -

^

1=3 1=1 n—ln—l

1=3 q=l n—ln—l

J]5]r«- 1Vi,|!!II|.lj.

(4.25)

1=3 1=1

By Theorem 2.3 we have, whenever t\ < t n __i, i,

n-l

n-1 1

9 =Z

q=l q=l

II ^ a " - 9 - 1 ^ ! ! = || ^ i i " - 9 - 1 ^ - u ) | | = ||/ - un-l\\ < c, and n-l

n - ll

IIH"-9-1! < ct n _ t < c.

Using the last three estimates for an estimation on the basis of (4.25), it follows that for all tj < tn, n-l

/

||2fln-lj|| < C + C ^ 1=3 \

_

n-l

\\^ n_U\\+kJ2tn-qWi,l-Vn-l, 1=1

Finally, applying Lemma Lemma 4.1 4.1 to this inequality and taking taking (4.22) (4.22) into into account account gives us (4.24). For the considered class of equations it is worth studying the perturbed problem (4.2) even if 2l n = 21 is independent of tn. The following result is applicable just in this case.

78

CHAPTER CHAPTER 4. 4. EQUATIONS EQUATIONS WITH WITH MEMORY MEMORY

T h e o r e m 4.4. Let the the conditions conditions of of Theorem Theorem 4-3 4-3 be be satisfied satisfied and and let, let, with with a seminorm ||| |||*o;n that forms forms aa A^(*; A^(*;7*0)-concordant 7*0)-concordant pair pair with with 21 21for for some some 7*0 € [0,1) and $0 G (0,1], the operator 2$ n be under the condition, condition, for for all all k e (Q,K], t n e Qk, and x G X*, ;n.

(4.26) (4.26)

Then the solution Y n of problem (4-2) satisfies satisfies the the stability stability estimate estimate (4-23). (4-23). Proof. The claim can be shown shown by by using, using, in in fact, fact, the the same same argument argument as as inin the proof of the preceding preceding theorem. theorem. First of all, applying (4.26), (4.26), letting letting formally formally 77 == 71 71 == 7* 7* — —70 70 == 00 and and tf — °d\ — 1, and leaving leaving 7*0 7*0 and and i?o i?o just just the the same same as as in in the the statement statement ofof the the present theorem, with the the aid aid of of Theorem Theorem 2.12 2.12 itit follows follows that that for for all all tjtj 1, with smooth boundary, and let let A(t) A(t) be be an an elliptic elliptic partial partial differential differential operator operatorofof second order defined by

with Dirichlet boundary conditions, conditions, where where the the coefficients coefficients a,ji(t, x) = aij(t, x) are smooth and real-valued real-valued on on [0, [0,T] T] xx C1J?. C1J?. The The operator operator .A(t) .A(t) isis assumed assumed to be uniformly elliptic in in the the following following standard standard sense, sense, with with some someao ao >> 0,0,

> ao ]T) Mj

for

all /ij G M, j = 1,..., d.

Then, thinking of .A(i) as as aa closed closed linear linear operator operator in in Loo(^) Loo(^) with with suitably suitably chosen domain, it can be be shown shown that that A(t) A(t) G G <S( max{£, max{£,1}. 1}. Then, for any fixed integer m\ m\ > > £, £, the the pairs pairs (|(| \\t; A(t)) and (| |* ;t; A(t)) are (^\m\\ip,a)- and {*;£ {*;£t\m\\i(z) satisfies (5.12). Since SinceA(t) A(t)£ £S(ip,a) S(ip,a)and andthethe pair ;A(i)) | (£|m|l,

and

K ( z ) | < Ce cn|z|~* /or aZZ z g Int23(di).

Then, with some K > 0 sufficiently small, the operator operatoru(—kA(t)) u(—kA(t)) satisfies satisfies the estimate (5.18) for all t G [0,T], fc G (0,X], and n G N U {0} sucfr i/iat nA; < C. Furthermore, the condition condition nk < C is not needed if a > 0. Proof. Taking = {z G C : \z\ = d/(2n + 2) + Rk,z~1 = j " 1 ^ '

for 1 < j < m and 1 < I < u;

s=l

and V

T>{m)

Y^ bs4~1(lsi

= J^biO-

-4)

iorl, are specified by the symplifying assumptions B(^) and C(z/). It is known (see, e.g., [53, 65, 81]) that all methods of this family are ^4-stable and, for any fixed v > 1, the i^-stage Gauss-Legendre method method is of order 2v and of stage order v.l& Also, it follows from Saff & Varga [137] that the eigenvalues of the coefficient matrix a are localized in the set IntS^/2- It is pointed out in Dekker & Verwer [65] that the matrix a is non-singular so that (5.24) and (5.25) hold. Apart from everything else, it is not hard to see that r(oo) = 1 if v is even and r(oo) = — 1 if v is odd. It is therefore clear from the above reasonings that Gauss-Legendre method method is of type Cjj(ip) for if v is even, the corresponding Gauss-Legendre any fixed

{v). For the i^-stage Radau IIA method, Cj, j = 1 , . . . , u, are the roots of the equation Pv-i(l

- 2c) + Pu(l - 2c) = 0,

while the other coefficients are specified by the simplifying conditions S(z^) S(z^) and G(u). It is known (see, e.g., [53, 65, 81]) 81]) that the methods of both families are A-stable, all of which satisfy the condition r(oo) = 0 or, equivalently, a(oo) = 1, and, for any v > 1, the i/-stage Radau IA or IIA method is of order 2/^ — 1. At the same time the stage order is v — 1 for Radau IA methods and v for Radau IIA ones. Also, by Saff and Varga [137] the eigenvalues of the coefficient matrix a are localized in I n t S ^ - Moreover, the matrix a itself is non-singular (see, e.g., [65]) so that (5.24) and (5.25) hold. Therefore any Radau IA or IIA method is both of type Vi(ip) and of type Vj(tp) for any fixed if G (0,7r/2). In particular, the widely popular backward backward Euler Euler method method is nothing but the 1-stage Radau IIA method with the Butcher array

/ 1 1\ Further, we discuss the behaviour of the Lobatto IIIA, IIIB, and IIIC methods (see, e.g., [53, 65, 81]). For all these methods, c\ = 0, c v = 1 and Cj, j = 1 , . . . , v, are the roots of the equation

5.2. DISCRETIZATION DISCRETIZATION

101

while the other coefficients coefficients are are specified specified by by the the simplifying simplifying assumptions assumptions rB(u) and Q{v) for Lobatto IIIA methods, by ®(i/) and V(v) for Lobatto IIIB methods, and by 'B(^) and and Q(v Q(v — 1) and the additional condition condition

81]) that that all all these these for Lobatto IIIC methods. methods. It is known (see, e.g., [53, 65, 81]) methods are A-stable and and the the order order of a f-stage method is always 2u — 2. For the ^-stage Lobatto IIIA IIIA and and IIIB IIIB methods methods the the stage stage order order is v and v — 2, respectively, and the stability stability function function r(z) r(z) — —11 — a (z) coincides with matrix a is the (v — l,v — 1)-Pade approximation to e2 but the coefficient matrix singular. Nevertheless, it follows from [65] that deg[ det (i — a)] = v - 1,

which shows that deg[ £ ( p ) . We first consider the case case with with aa constant constant operator, operator, that that is, is, where where A(t) A(t) is independent of t. (We (We therefore therefore write write AA instead instead of of A(t).) A(t).) Let Let the the RK RK method be of type C((p) C((p) for for some some (p (pG G (0, (0, TT/2). Then, for any fixed a G G M, M, ifif K > 0 is sufficiently small, small, the the functions functions a(—k\) a(—k\) and and Wj(-kX), Wj(-kX), jj == 11, ,. . . ., , v,v, are denned for any k € (0, (0, K] K] and and A AG G (Int (Int S^, S^,U U{0}) {0}) ++ ex, and one can take K — oo if a > 0. It is then then not not hard hard to to see see that that in in the the scalar scalar case case A(i) A(i) == XI, XI, problem (5.19), (5.20) reduces reduces to to the the following following one one

In the operator case A A G G S(tp,a), S(tp,a), dealing dealing with with (5.19), (5.19), (5.20), (5.20), one one should should ,^, from (5.19) and (5.20), (5.20), that that is, is, solve solve (5.20) (5.20) for for eliminate Y nj, J' — lj Ynj and insert them into into (5.19). (5.19). This This procedure procedure certainly certainly isis realized realized ifif the the system of equations (5.20) (5.20) is is uniquely uniquely solvable. solvable. ItIt may may however however happen happen that that (5.20) is not uniquely solvable solvable while while the the operators operators r(—kA) r(—kA) and and Wj(-kA), Wj(-kA), j — 1 , . . . , v, are nevertheless nevertheless defined defined and and bounded bounded on on X. X. In In particular, particular, itit follows from Lemma 5.3 5.3 that that the the definedness definedness and and boundedness boundedness ofof r(—kA) r(—kA) and of Wj(-kA), j = 11,,......,, v, v, are are observed observed for for all all kk GG (0, (0,K], K], with with KK >> 00 sufficiently small, if A G G S(ip, S(ip, a) a) and and the the RK RK method method isis of of type type C(3t specified specified by the rereDefinition 5.10. The The discrete discretefunction functionYn : Q/- — Definition 5.10. currence (5.30) (5.30) is called called the the extended extended solution solution of the Runge-Kutta Runge-Kuttaprocedure procedure currence autonomous case. case. (5.19) and and (5.20) (5.20) in the autonomous (5.19) Further, we weturn turn to tothe thegeneral generalcase casewhere wherethe theoperator operator A(t)may may depend Further, A(t) depend convenience we weuse use on t. For convenience diag (A(tm),.. (A(tm),.. .,A(t .,A(tni/)), = diag 1

V(t)= =diag V(t)

= {9jl(-kA(t)) (-kA(t))vxv,

and a{-kA(t)). &(t) = k~la{-kA(t)).

(5.31) (5.31)

what follows follows we weconsider consideroperator operatorequations equationsof the form, form, given given u = In what

(«i,..,«,felM, («i,..,«,fel (i + kaDn)z = ou,

(5.32) (5.32)

(i + kaT)n)z = e 0 0 u,

(5.33) (5.33)

form, given given u E X, or of the form,

5.3. CONNECTION CONNECTION WITH DISCRETE EVOLUTION EQUATIONS 105 5.3. WITH DISCRETE EVOLUTION EQUATIONS 105 which aresolved solved for for zz == {z\,...,zj) {z\,...,zj) which are

Vom Vorn. x equation GGVom x x x Vorn. IfIf equation V V

(5.32) (equation (5.33)) isuniquely uniquelysolvable solvable forany any GX^ X^ (for (for any u G (5.32) (equation (5.33)) is for u uG any u G X), we wedenote denotethe thecorresponding corresponding solution operator by(i(i++ka karD Dn)~ )~1a (by (by X), solution operator by -1 (i + + ka1) ka1)n)~ e) so sothat that zz==(i(i++ka ka(D Dn)~ )~lau. au. (z (z==(t(t++fcaD fcaDn)-1 u). (i )~le) e >u). Now we wepresent presentsufficient sufficient conditions forproblem problem(5.19), (5.19), (5.20) tobe be Now conditions for (5.20) to equivalently representable inthe theform form (1.1). (1.1).In Inwhat whatfollows follows different equivalently representable in wewe useuse different formsof ofHolder-continuity Holder-continuity ofA(t) A(t)and and this main mainidea idea separates thebelow below forms of this separates the assertions from each other. We start, however, bygiving givingan anauxiliary auxiliaryresult result by assertions from each other. We start, however, bein incommon commonuse usefor forthe the corresponding proofs. whichwill willbe which corresponding proofs. Lemma5.7. 5.7.Let Let A(t) S(ip, G a) a) and andlet let the methodbe beof oftype type V(00 or K sufficiently small, forall allkk G G(0, (0,K] K]and and G[0, [0,T], T],the theoperators operators and matrix sufficiently small, for tG and matrix l operators a(-kA(t)),p{-kA{t)), p{-kA{t)), v(-kA(t)), w(-kA(t)), kaV(t))~ (i(i++ kaV(t))~ , operators a(-kA(t)), v(-kA(t)), w(-kA(t)), M 1 M (i++ka'D(t))ka'D(t))-a, kT)(t)(i (i++kaV^y^e kaV^y^e areare defined onXXor oron on kV(t) (i kV(t) a, kT)(t) defined on X X, with with \\a(-kA{t))\\+ +\\p(-kA(t))\\ \\p(-kA(t))\\+ +\H-kA(t))\\ \H-kA(t))\\ + +Hw(-fcA(t))|| Hw(-fcA(t))|| bounded on on X. X. Now Now kaT>n)~1a are defined and bounded a simple estimation of the the expression expression (i(i ++ Sn)~ Sn)~1SnU leads, with the aid of of (5.42), (5.47), and (5.48), to the the desired desired estimate estimate for for the the first first term term on on the the left-hand side of (5.43), and and the the same same for for the the second second term term follows follows by byusing usingaa similar argument. It is also also seen seen from from the the above abovereasonings reasoningsthat that the theentries entriesofof the operator matrices kT)n{\ + kaT> n)~la and kT) n(i + kaT) n)~1e are defined and bounded on X. It remains to show (5.44) and and (5.45). (5.45). We We denote denote 55n = k(T> n-T>(tn)) (i+ l ka'Dn)- a. It follows from (5.46) that that (i + kaVn)-^

= -(i + ka'D{t ka'D{tn))-1aBn + (i +

fcoD^n))-^.

(5.49) (5.49)

Multiplying now both sides sides of of this this equality equality from from the the left left by by kT>(t kT>(tn) and using the identity kDitn) (i + kaT){t n))-la = i - (i + / c a D ^ ) ) " 1 ^ which can easily be verified verified due due to to the the fact fact that that aa and and (i(i++ ka.T)(t ka.T)(tn))~l are permutable, we find that

Inserting this result into the the right-hand right-hand side side of of the the identity identity

kVn{i + fcoDn)-^ - kV(t n) (i + kaV n)-1a+

S n,

leads immediately to (5.44). (5.44). Further, acting similarly to to the the manner manner leading leading to to (5.50) (5.50) and and (5.44), (5.44),we we get kVnii + kaDn)-^

= kV(t kV(tn) (i + kaV(t n))-1e + (i + fccd)^))" 1 xfc(D n -D(t n ))(i + fca2) n)-1e (5.50)

and (5.51) Thus we see that (5.45) follows follows when when inserting inserting (5.51) (5.51) into into the the right-hand right-hand side of (5.50) and using (5.48) (5.48) and and the the trivial trivial identity identity Now the proof of the lemma lemma isis complete. complete.

5.3. CONNECTION

WITH WITH DISCRETE DISCRETE EVOLUTION EVOLUTION EQUATIONS EQUATIONS 109 109

Y n G X{u) be denned by Proof of Theorem 5.1. Letting Y )

I n—

we rewrite problem (5.19), (5.19), (5.20) (5.20) as as follows follows Yn+l =Yn- kbTT>nYn + kbT$n

for tn G Clk,

Yo = 9fy°

(5.52)

fca$ n.

(5.53)

and Yn = e®y n-fcaD nYn +

Since, by Lemma 5.8, the the entries entries of of the the operator operator matrices matrices (i(i++ ka karDn)~1a and 1 (i + feoDn)~ e are defined on X for /c G (0,-ftT] and t n G fifc, it follows from (5.53) that Y n = (i + kaVn)-^

® y n +fc(i+

fcaD,,)-^^.

(5.54) (5.54)

Inserting further (5.54) (5.54) into into (5.52), (5.52), we we find find that that problem problem (5.19), (5.19), (5.20) (5.20) can can be rewritten in the equivalent form

Yn+l = (l-kbTVn{i

1

)

T

+kb (i - kVn(i °

(5.55) (5.55)

the above above Therefore, to show the claim, it suffices to combine (5.55) with the representations (5.44) and and (5.45), (5.45), noting noting that that bbT'D(t)(i + ka'D(t))' 1e = 2l(t) D D and b T (i + kaV{t))- 1 = w T ( - i t ^ ( t ) ) . 2 5

R e m a r k 5.1. In particular, the assertion of Theorem 5.1 remains valid valid if, instead of holfLfi?; | \t], it is based on either of the more restrictive hypotheses hol®[i9; | | t ] or hol[i?]. So we have shown that under under some some reasonable reasonable conditions conditions problem problem (5.19), (5.19), (5.20) can be equivalently represented represented in the form (1.1), that is, as the Cauchy problem for an evolution equation in discrete time. In particular, this result yields that the then the unique unique solvability solvability of problem (5.19), (5.20) then convergence only. only. We We conconfollows and it remains to examine stability and convergence centrate on these questions questions in subsequent chapters of the the book. book. 25

Note that, as follows from from Lemma Lemma 5.8, 5.8, the the operators operators 2l(£ 2l(£n), 58n, 2ln are bounded on on X for k € (0,K] and tn 6 Clk.

110 CHAPTER CHAPTER 5. DISCRETIZATION BYRUNGE-KUTTA RUNGE-KUTTA METHODS 110 5. DISCRETIZATION BY METHODS

5.4 Comments bibliographical 5.4 Comments and and bibliographical remarks remarks Section The proofof ofLemma Lemma5.2 5.2follows followsthat that ofLemma Lemma2.2 2.2ininour our Section 55..11: :The proof of paper[37]. [37]. paper Section5.2: 5.2:For For generalbooks books inthe the field of ofRK RK methods,we werefer referthe the Section general in field methods, reader toButcher Butcher[53], [53], Dekker &Verwer Verwer[65], [65], Hairer, N0rsett, andWanWanreader to Dekker & Hairer, N0rsett, and ner [80], [80], Hairer& &Wanner Wanner[81], [81], and Stetter[155]. [155]. Although Althoughwritten written in ner Hairer and Stetter in the context ofordinary ordinarydifferential differential equations, these manuals wide present the context of equations, these manuals present aa wide rangeof ofideas, ideas,notions, notions, terminology. Moreover, in[53, [53,65, 65,81, 81, 155], range terminology. Moreover, in 155], theythey pay pay attention tothe the analysisof ofdiscretizations discretizations ofstiff stiffproblems. problems. mention analysis of We We also also mention attention to recentbook book ofHundsdorfer Hundsdorfer &Verwer Verwer[86] [86] which discretizations a recent of & which dealsdeals with with discretizations of both bothODEs ODEs andPDEs. PDEs. The The bookscited cited above aretherefore thereforerelated related to of and books above are to the present present which isintended, intended,in inparticular, particular, forapplications applications inorder orderto to the oneone which is for in examine discrete versions ofpartial partialdifferential differential equations. of Most the modmodexamine discrete versions of equations. Most of the ern approaches approaches tothe the studyof ofdiscretization discretization methods deal abstract with abstract to study methods deal with ern andthe the situationof ofpartial partialdifferential differential equations can differentialequations, equations, differential and situation equations can thenbe besettled settledas asaaparticular particular case. Inthe the abstract abstract case however, then case. In case theythey meet,meet, however, new features features ofwhich whichthe thecentral centralone oneisisthe the problemof ofinfinite infinitedimension. dimension. new of problem As concerns concerns different aspects ofgeneral generalRK RKmethods methodsconsidered considered inHilbert Hilbert As different aspects of in and Banach Banach space settings andapplied appliedto tolinear, linear,quasilinear, quasilinear, andfully fully nonnonand and and space settings linearproblems, problems, wemention mentionour ourwork work[16, [16, 19,20, 20, 22, 27, 30,36] 35, 36] linear we 19, 22, 25,25, 26,26, 27, 30, 35, and the thework workof ofBeyn Beyn& &Garay Garay[47], [47], Gonzalez &Ostermann Ostermann[71], [71], Gonzalez and Gonzalez & Gonzalez et al.[72], [72],Gonzalez Gonzalez& &Palencia Palencia[73, [73, 74,75], 75], Gudovich& &Terteryan Terteryan [78], et al. 74, Gudovich [78], Karakashian [90], Keeling [92], Lubich &Ostermann Ostermann[105, [105, 106,107], 107], NakNakKarakashian [90], Keeling [92], Lubich & 106, aguchi& &Yagi Yagi [115], [115],Ostermann Ostermann &Roche Roche[123], [123],and andOstermann Ostermann &ThaiThaiaguchi & & hammer[125]. [125]. hammer Section5.3: 5.3: Ourpresentation presentation is,in infact, fact, an animproved improvedversion version ofour our Section Our herehere is, of earlier work (These arealso alsoannounced announced in[16, [16,22, 22, 26].) 26].) earlier work [19,[19, 35]. 35]. (These results results are in

Chapter 6

Analysis of Stability In this chapter we further investigate the Runge-Kutta discretizations discretizations discussed previously in Chapter 5. We concentrate here on examining stability for discrete problems of the form (5.19), (5.20) by using seminorms | \t that form concordant pairs pairs with with A(t) in the sense of Definition 5.2 above. It has already been pointed out before that our techniques are based on reducing the starting problem (5.19), (5.20) (5.20) to the discrete Cauchy problem (1.1). In fact, under the assumption that A(t) is (, u)-sectorial with some

0, andrn G N such thatm—l < £ < m. Let also the RK method be A j (if)-stable and let a rational function u(z) be ((p)-regular and subject to deg[cj) < —m. Then there exist a set T of Ai-configuration and numbers 5 > 0 and K > 0 such that for all k G (0,K), t G [0,T], A £ Int (TUV(8k)), andu£X, \u(-kA(t)) (XI - k%{t))k%{t))-mu\t < C|A|*- m|M|.

(6.1)

In the case £ = m — 1, under the extra requirement that the pair (| \t;A(t)) is (£|£|, 1 and \u\t < C\\u\\ for all t € [0,T] and u G GX X ifif ££ == 0, 0,

^^ '' ''

it is enough to suppose that deg[o;] < —£ instead of < —m. If a > 0 in in addition, then the above still holds for 5 = 0 and K = oo. Proof. Assuming first that (6.2) is not generally satisfied, we then act under the condition deg[u;] < —m. Let 0 be the same as in the proof of Lemma 5.2. Observe that since the method is A(00be bechosen chosensufficiently sufficiently small sothat that p(z), p(z),a(z), a(z),and and ui{z)have haveno nopoles poles d> small so ui{z) V(d). Also, Also,we wechoose chooseK K< 0>and K =Koo=if oo R= in By the theaccepted acceptedconditions conditions there exists setT' T'ofof Ai-configuration such By there exists aa set Ai-configuration such that the thefunction functiona(—z) a(—z) maps intoT'. T'. Let LetTTbebe a set Ai-configuration of that maps T,V1 into a set of Ai-configuration V1T, such thatT' T' \\ {0} {0}CC IntT. We Weselect select aa fixed fixeddo do >> 00sufficiently sufficientlysmall small to such that IntT. to have have do\p(z)\< < ^^ do\p(z)\

for X>(d). e for allall z ez X>(d).

(6.3) (6.3)

Let nowF\ F\bebethe the contourcoinciding coinciding the boundary boundaryof of the theset set Let now contour withwith the V(min{do\X\, d})US^. We Wethen then apply applythe theoperator operatorcalculus calculus formula (A.10) V(min{do\X\, d})US^. formula (A.10) 1 with kA(t) kA(t)ininplace placeof of££ and andwith withT\ T\ in inplace placeof ofH# H#+e toget, get,with with5 5 ==dQ dQ R, with R, +e to for all allkkGG(0, (0, K], 0 Int U (T V(6k)), and anduu€€X,X, for K], A 0AInt (T U V(6k)), m m m m G :=u(-kA(t)) u(-kA(t))((XI ((XI - a(-kA(t)))(A- -a(oo))a(oo))J) u G := - a(-kA(t)))-- (A J) 1 = 7r^ 7r^ [ [ am a(X;z)u:(-z)(zI-kA(t))(X;z)u:(-z)(zI-kA(t))udz, = udz, m

(6.4) (6.4)

where where m m (A; z) z) == (A (A- -aa((--zz) ) — ) —-- (A (A- - a(oo))" a(oo))" . 5m m(A;

Integratingby byparts partson onthe theright-hand right-handside side of (6.4), (6.4),we weget getinstead instead Integrating of G G

m m yym (X;z)(zI-kA(t))udz, udz, m(X;z)(zI-kA(t))-

(6.5) (6.5)

where where

{

am (X;z)uj(-z) m(X;z)uj(-z)

if mif= 1, m = 1,

m 2 m (A;z1 )a;(-zi)^ )a;(-zi)^ (m m--11) ) / / (z-2i) (z-2i) - 5m ifif mm>>2,2, m(A;z 1 tv>1 tv>1 while thepath path of of integration integrationfrom from ooe~ to zz lies lies on on Y\. Y\. while the ooe~ to Assumefor for aamoment momentthat thatwe wehave have shown shownthe theestimate, estimate,for for A A^^Int Int Assume TT and GT\, T\, and zz G

IAMzl)IAMzl)-1

ifif mm= =l , l ,

((

..

( tt66))

Sincethe Since thepair pair (| (| \t;A(t)) ;A(t)) is is (^|m|> dd dd0 1 , we derive, using (6.5), (6.6), (6.7), and the fact that that 11 £(ZJ + 1) > m, we 6

Clearly, (6.13) and (6.16) (6.16) hold hold for for methods methods of of class class SII SII as as well. well.

6.1. SOME RESOLVENT ESTIMATES ESTIMATES

119

have, 7 for z € V\ n dT,(pi, \z\ < min {l,

, Qz < 0,

|6m(A;z)| < C J"

< c?u

,m-2

Jz

r,wm+m—2— |A|~ |A|~1//ro, c

< c\x\~

m

(x-\z\)

m-2

dx

J\z\

o

\

x

dx

l

-\

(6.27)

With the aid of (6.26) and and (6.27) (6.27) we we then then obtain, obtain, for for all all AA££ Int Int (TuV(6k)), (TuV(6k)),

\G\t < C\\u\\

which leads to (6.20), since since the the second second term term of of the the sum sum on on the the right-hand right-hand 8 side is dominated by the first first one, one, in in view view of of (6.19). (6.19). Let now m > 2 and m — l/{w + 1) < £ < m so that that ?? — {w + l)m. It is then not hard to see that for for zz G GT\, T\, \z\ \z\ < < min min {l, {l, , the second term 7 8

The second integral in the the second second line line does does not not enter enter ifif |A| |A| >> 1.1. It is easily seen that \G\t \G\t is is bounded bounded as as stated stated for for |A| |A| >> d^n7^^tD

as well.

120

CHAPTER6.6.ANALYSIS ANALYSISOFOFSTABILITY STABILITY CHAPTER

in the the second second line line and and the theterm termininthe thethird thirdline lineof of(6.26) (6.26) bounded areare bounded by by C and and C|A| C|A|x/ro, respectively, respectively, and, and, thus, thus,both bothadmit admita acommon commonupper upperbound bound C^l"1 . Therefore, Therefore, instead instead ofof (6.26) (6.26)we wehave have|6|6 (A;z)| 0 sufficiently small, the operators (kA(t) — 2;/)" defined on X and uniformly bounded with with respect respect to k G (0, K] and t G [0, T], and, clearly, the same is true for the operator P(kA(t)). representation for the difference 21 (t) — Below we need to get a suitable representation 2l(s). To that end, we put g(t) = p(-kA(t)) and ?(t) = P(kA(t)) and note that using (6.39) leads to the identity

which gives - 2l(s) =

(bik2i(s) + bQg{s) boQ(t) + 7(t)g(t))

=:

Q1 + Q2.

(6.40) (6.40)

Further, for convenience' sake we put (j>i{t) = (kA(t) — zrf)'1, gij(t) — (