Numerical Linear and Applications (No. 1, 2008)

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2008; 15:1–11 Published online 24 October 2007 in...

18 downloads 472 Views 912KB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2008; 15:1–11 Published online 24 October 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nla.560

Perturbation analysis for the generalized Schur complement of a positive semi-definite matrix Musheng Wei1, ∗, † and Minghui Wang1, 2 1 Department

of Mathematics, East China Normal University, Shanghai 200062, China of Mathematics, Qufu Normal University, Shandong 273165, China

2 Department

SUMMARY Let

P=

A

B

BH

C

0

and S = C − B H A† B be the generalized Schur complement of A0 in P. In this paper, some perturbation bounds of S are presented which generalize the result of Stewart (Technical Report TR-95-38, University of Maryland, 1995) and enrich the perturbation theory for the Schur complement. Also, an error estimate for the smallest perturbation of C, which lowers the rank of P, is discussed. Copyright q 2007 John Wiley & Sons, Ltd. Received 3 April 2007; Revised 17 September 2007; Accepted 19 September 2007 KEY WORDS:

generalized Schur complement; SVD; Moor–Penrose inverse

1. INTRODUCTION In this paper, we use the following notation. Cm×n is the set of m ×n matrices of complex entries. For any matrix A ∈ Cm×n , rank(A) is the rank of A, R(A) is the range of A, AH is the conjugate transpose of A, A† is the Moore–Penrose inverse of A. A0 (>0) implies that A is a Hermitian positive definite (semi-definite) matrix. · = ·2 is the Euclidian vector norm or the spectral matrix norm. ∗ Correspondence †

to: Musheng Wei, Department of Mathematics, East China Normal University, Shanghai 200062, China. E-mail: [email protected]

Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 10771073 Contract/grant sponsor: Science and Technology Commission of Shanghai Municipality; contract/grant number: 04JC14031

Copyright q

2007 John Wiley & Sons, Ltd.

2

M. WEI AND M. WANG

For a given block matrix

A P= C

B D

with A invertible, the Schur complement of A in P is S = D −CA−1 B. When A is not invertible, then the generalized Schur complement of A in P is S = D −CA† B. The Schur complement is a useful tool in matrix analysis. A great deal of work on this topic has been done in the literature, see [1–10] and the references cited therein. The perturbation analysis of the Schur complement has been deduced in the literature [11]. When the matrix P is positive semi-definite, Stewart [12] established a sharper bound that is given in the following theorem. Theorem 1.1 Suppose that the matrices A ∈ Cn 1 ×n 1 , C ∈ Cn 2 ×n 2 , B ∈ Cn 1 ×n 2 are given, and A B P= 0 BH C A>0. Let

Pˆ =

A +A B H +B H

Aˆ ≡ C +C Bˆ H B +B

Bˆ Cˆ

be the perturbed version of P with A = AH , and AA,

BB,

CC

where >0. Denote = AA−1 . If 0

from |t j − tˆj | ( j = 1, . . . , n 2 ), and from (24), T2 W2H Wˆ 1 Tˆ1−1 − W2H Wˆ 1 T2 W2H Wˆ 1 − W2H Wˆ 1 Tˆ1 Tˆ1−1 ¯/tˆp Therefore, one obtains W2H Wˆ 1 ¯/tˆp +T2 W2H Wˆ 1 Tˆ1−1

(25)

From (25), one obtains W2H Wˆ 1

¯ ¯ tˆp −t p+1 t p −t p+1 − t p −t p+1 −

(26)

Similarly, one obtains W1H Wˆ 2

¯ t p −t p+1 −

t p −t p+1 −

(27)

Then from (23)–(27), one obtains ˆ ¯ +max{T2 W2H Wˆ 1 , W1H Wˆ 2 Tˆ2 } C −C ¯ max{T2 , Tˆ2 } ¯ + t p −t p+1 − t p+1 + ¯ 1+ t p −t p+1 − t p+1 + 1+ t p −t p+1 − One then obtains the desired estimate in (21). Copyright q


Numer. Linear Algebra Appl. 2008; 15:1–11 DOI: 10.1002/nla

10

M. WEI AND M. WANG

Remark 4.2 ˆ is small, the quantity |t j − tˆj |S − S ˆ for j = 1, . . . , n 2 , that is, the perturbations of When S − S ˆ could be large. However, if t p −t p+1 , the the eigenvalues of S are also small. Note that C −C ˆ ˆ is gap between the eigenvalues, is large enough such that S − S0. In this paper, we generalize the analysis to the case in which P0 and A0. The derived bounds are sharper than those in [12]. We also discuss an error estimate for the smallest perturbation of C which lowers the rank of P.

ACKNOWLEDGEMENTS

We are grateful to the editor and the anonymous referee for providing many useful comments and suggestions, which improved the presentation of the article.

REFERENCES 1. Carlson D, Haynsworth E, Markham T. A generalization of the Schur complement by means of the Moore–Penrose inverse. SIAM Journal on Applied Mathematics 1974; 26:169–175. 2. Cottle RW. Manifestation of the Schur complement. Linear Algebra and its Applications 1974; 8:189–211. 3. Ando T. Generalized Schur complements. Linear Algebra and its Applications 1979; 27:173–186. 4. Styan GPH. Schur complements and linear statistical models. In Proceedings of the First International Tampere Seminar on Linear Statistical Models and their Applications, Tampere, Finland, August–September 1983, Puntanen S, Pukkila T (eds). Department of Mathematical Sciences, University of Timpere, 1985; 37–75. 5. Bapat RB. A refinement of Oppenheim’s inequality. In Current Trend in Matrix Theory: Proceedings of the Third Auburn Matrix Theory Conference, Auburn University, Auburn, AL, March 1986, Uhlig F, Grone R (eds). North-Holland: New York, 1987; 29–32. 6. Butler CA, Morley TD. Six generalized Schur complements. Linear Algebra and its Applications 1988; 106: 259–269. 7. Smith R. Some interlacing properties of the Schur complement of a Hermitian matrix. Linear Algebra and its Applications 1992; 177:137–144. 8. Wang B, Zhang F. Schur complements and matrix inequalities of Hadamard products. Linear and Multilinear Algebra 1997; 43:315–326. 9. Redivo-Zaglia M. Pseudo-Schur complements and their properties. Applied Numerical Mathematics 2004; 50: 511–519. 10. Zhang F. The Schur Complement and its Applications. Springer: Berlin, 2005. 11. Higham NJ. Accuracy and Stability of Numerical Algorithms (2nd edn). SIAM: Philadelphia, PA, 2002. 12. Stewart GW. On the perturbation of Schur complement in positive semidefinite matrix. Technical Report TR-95-38, University of Maryland, 1995. 13. Albert A. Condition for positive and nonnegative definite in terms of pseudoinverse. SIAM Journal on Applied Mathematics 1969; 17:434–440. 14. Horn RA, Johnson CR. Topics in Matrix Analysis. Cambridge University Press: Cambridge, U.K., 1988. 15. Stewart GW, Sun J-G. Matrix Perturbation Theory. Academic Press: Boston, 1990. 16. Marsaglia G, Styan GPH. Equalities and inequalities for ranks of matrices. Linear and Multilinear Algebra 1974; 2:269–292. Copyright q



PERTURBATION ANALYSIS FOR THE GENERALIZED SCHUR COMPLEMENT

11

17. Golub GH, Van Loan CF. Matrix Computations (3rd edn). Johns Hopkins University Press: Baltimore, MD, 1996. 18. Wei M. Perturbation theory for the Eckart–Young–Mirsky theorem and the constrained total least squares problem. Linear Algebra and its Applications 1998; 280:267–287. 19. Demmel JW. The smallest perturbation of a submatrix which lowers the rank and a constrained total least squares problem. SIAM Journal on Numerical Analysis 1987; 24:199–206.

Copyright q



NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2008; 15:13–34 Published online 22 November 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nla.561

A comparative study of efficient iterative solvers for generalized Stokes equations Maxim Larin∗, † and Arnold Reusken Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany

SUMMARY We consider a generalized Stokes equation with problem parameters 0 (size of the reaction term) and >0 (size of the diffusion term). We apply a standard finite element method for discretization. The main topic of the paper is a study of efficient iterative solvers for the resulting discrete saddle point problem. We investigate a coupled multigrid method with Braess–Sarazin and Vanka-type smoothers, a preconditioned MINRES method and an inexact Uzawa method. We present a comparative study of these methods. An important issue is the dependence of the rate of convergence of these methods on the mesh size parameter and on the problem parameters and . We give an overview of the main theoretical convergence results known for these methods. For a three-dimensional problem, discretized by the Hood–Taylor P2 –P1 pair, we give results of numerical experiments. Copyright q 2007 John Wiley & Sons, Ltd. Received 6 February 2007; Revised 17 October 2007; Accepted 17 October 2007 KEY WORDS:

generalized Stokes problem; preconditioned MINRES; inexact Uzawa method; multigrid methods; Vanka and Braess–Sarazin smoothers

1. INTRODUCTION Let ⊂ R3 be a bounded polygonal domain with a Lipschitz boundary = *. We consider the following generalized Stokes problem. Given f, find a velocity u and a pressure p such that u−u+∇ p = f

in

∇ ·u = 0

in

u=0

on

∗ Correspondence

(1)

to: Maxim Larin, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany. † E-mail: [email protected] Contract/grant sponsor: German Research Foundation (DFG); contract/grant number: SFB 540

Copyright q


14

M. LARIN AND A. REUSKEN

The parameters >0 (viscosity) and 0 are given. Often the latter is proportional to the inverse of the time step in an implicit time integration method applied to a nonstationary Stokes problem. Note that this general setting includes the classical (stationary) Stokes problem ( = 0). This problem is discretized on a tetrahedral grid with a pair of conforming finite element spaces that is inf–sup stable. In our experiments we use the P2 –P1 Hood–Taylor pair. The resulting discrete problem is of saddle point type with a symmetric indefinite matrix. In this paper we study efficient iterative solvers for this linear system. In particular, the efficiency (robustness) of the solvers with respect to variation in the mesh size parameter and in the problem parameters and is studied. We consider a multigrid method with Vanka and Braess–Sarazin smoothers, a preconditioned MINRES method and an inexact Uzawa method. In the latter two methods multigrid preconditioners are used for the scalar problems for each velocity component. A comparative study of the preconditioned MINRES and inexact UZAWA method with other preconditioned Krylov subspace methods for this problem class is given in [1]. Numerical studies on the performance of coupled multigrid problems for (generalized) Stokes and Navier–Stokes can be found in, for example [2–6]. We do not know of any literature in which a systematic comparison between coupled multigrid, preconditioned MINRES and inexact Uzawa type of methods for this class of generalized Stokes equations is given. In this paper, we present such a comparative study. We give an overview of the main theoretical results that are available for these methods. From this it follows that concerning theoretical convergence results the state of affairs is much better for the preconditioned MINRES and the inexact Uzawa method than for the coupled multigrid method. We pay special attention to the case = 0, >0 variable. In this case, variation of the parameter corresponds to a rescaling of the velocity unknowns. We show that for all methods considered here the rate of convergence is essentially independent of this rescaling. We also investigate the efficiency of the different methods by means of numerical experiments. It turns out that all methods show good robustness properties with respect to variation in the mesh size and in the parameters and . The paper is organized as follows. In Section 2 the weak formulation and the finite element discretization are given. In Section 3 the coupled multigrid method with Braess–Sarazin and Vanka smoothers is described. For the method with Braess–Sarazin smoother a convergence analysis known from the literature is presented in a different form. In Section 4 we discuss the preconditioned MINRES and inexact Uzawa methods. We recall known convergence results for these methods. A comparison of all these methods from a theoretical point of view is given in Section 5. In Section 6 a numerical study of these methods is presented and conclusions are drawn.

2. WEAK FORMULATION AND FINITE ELEMENT DISCRETIZATION The weak formulation of (1) is as follows. Given f ∈ L 2 ()3 , we seek u ∈ H01 ()3 and p ∈ L 20 () such that (u, v)+(∇u, ∇v)−(div v, p) = (f, v) for all v ∈ H01 ()3 (div u, q) = 0

for all q ∈ L 20 ()

(2)

Here (·, ·) denotes the L 2 scalar product. Copyright q



EFFICIENT SOLVERS FOR GENERALIZED STOKES EQUATIONS

15

For discretization of (2) we use a standard finite element approach. Based on a quasi-uniform family of nested tetrahedral grids T0 ⊂ T1 ⊂ · · · we use a sequence of nested finite element spaces (Vl−1 , Q l−1 ) ⊂ (Vl , Q l ),

l = 1, 2, . . .

The pair of spaces (Vl , Q l ), l0, is assumed to be stable. By h l we denote the mesh size parameter corresponding to Tl . We assume that h l−1 / h l is uniformly bounded in l. For the theoretical analysis we assume that the pair has the following approximation property: inf u−v1 + inf p −q L 2 ch l (u2 + p1 )

v∈Vl

q∈Q l

∀u ∈ (H 2 ()∩ H01 ())3 , p∈H 1 ()∩ L 20 ()

We use the notation ·k , k = 1, 2, for the norms in H k (). In our numerical experiments we use the Hood–Taylor P2 –P1 pair. The discrete problem is given by the Galerkin discretization of (2) with the pair (Vl , Q l ). We are interested in the solution of this discrete problem on a given finest discretization level l = L. To solve this discrete problem we introduce the standard nodal bases in these finite element spaces. The representation of the discrete problem on level l in these bases results in a linear saddle point problem of the form ul Al BlT Al xl = bl with Al = , xl = (3) pl Bl 0 The dimensions of the spaces Vl and Q l are denoted by nl and m l , respectively. The matrix Al ∈ Rnl ×nl is the discrete representation of the differential operator I − and is symmetric positive definite. Note that Al depends on the parameters and . The matrix Al depends on these parameters, too, and is symmetric and strongly indefinite. In the remainder of the paper we consider iterative solvers for system (3) on the finest level L. Remark 1 The matrix Al is singular. Below we always consider Al on the subspace Rnl ×1⊥ , where 1⊥ is ml the corresponding finite element functions ql ∈ Q l satisfy the subspace of R of vectors forn which l ×1⊥ → Rn l ×1⊥ is invertible. q dx = 0. The mapping A : R l l 3. A COUPLED MULTIGRID METHOD We consider a multigrid method for the coupled system in (3). Below we discuss the components of this multigrid solver. The grid transfer operations: For the prolongation and restriction of vectors (or corresponding finite element functions) between different levels we use the canonical operators. The prolongation between levels l −1 and l is given by PV 0 Pl = 0 PQ where the matrices PV : Rnl−1 → Rnl and PQ : Rm l−1 → Rm l are matrix representations of the embeddings Vl−1 ⊂ Vl (quadratic interpolation for P2 ) and Q l−1 ⊂ Q l (linear interpolation for P1 ), respectively. For the restriction operator Rl between levels l and l −1 we take the adjoint of Pl (w.r.t. a scaled Euclidean scalar product). Copyright q



16


Coarse-grid operators: In the multigrid solver for the problem on the finest level L we need operators on the coarser levels L −1, . . . , 0. We use the matrices Al in (3). These result from the finite element discretization method applied on level l. For these matrices the Galerkin relation Al−1 = Rl Al Pl holds. The smoothers: In this paper we consider two popular smoothers for Stokes type of problems, namely the Braess–Sarazin smoother and a Vanka-type smoother. These smoothers and their properties are discussed in the following two subsections. 3.1. Braess–Sarazin smoother This smoother is introduced in [7]. With Dl = diag(Al ) and a given >0 the smoothing iteration has the form ⎛ ⎞ ⎛ ⎞ −1 ⎧ ⎛ ( j) ⎞ ⎫ ( j+1) ( j) ⎨ Al B T ul ul fl ⎬ u Dl BlT l ⎝ ⎠=⎝ ⎠− ⎝ l ⎠− (4) ( j+1) ( j) ( j) ⎩ Bl 0 0 ⎭ Bl 0 pl pl pl Each iteration (4) requires the solution of the auxiliary problem ⎛ ( j) ⎞ rl uˆ l Dl BlT ⎠ =⎝ ( j) pˆ l Bl 0 Bl ul ( j)

( j)

(5)

( j)

with rl = Al ul + BlT pl −fl . From (5) one obtains ( j)

Bl uˆ l = Bl ul and hence, ( j+1)

Bl ul

( j)

= Bl (ul − uˆ l ) = 0 for all j0

(6)

Therefore, the Braess–Sarazin method can be considered as a smoother on the subspace of vectors that satisfy the constraint equation Bl ul = 0. Problem (5) can be reduced to a problem for the auxiliary pressure unknown pˆ l : ( j)

( j)

Z l pˆ l = Bl Dl−1 rl −Bl ul

(7)

where Z l = Bl Dl−1 BlT . Remark 2 The matrix Z l is similar to a discrete Laplace operator on the pressure space. In practice, system (7) is solved approximately using an efficient iterative solver [7–10]. Once pˆ l is known (approximately), an approximation for uˆ l can easily be determined from ( j) Dl uˆ l = rl − BlT pˆ l . The iteration matrix of the smoother (4) is denoted by Sl . For a two-grid method, with 1 pre- and 2 post-smoothing steps, applied to (3) the iteration matrix is given by −1 Rl Al )Sl1 Ml = Sl2 (I − Pl Al−1

(8)

We derive a convergence result for this multigrid method with the Braess–Sarazin smoother. A multigrid convergence analysis is given in [7]; however, only for the case where Dl is replaced Copyright q



17

EFFICIENT SOLVERS FOR GENERALIZED STOKES EQUATIONS ( j)

by the identity matrix. In that paper the reduction of the velocity error ul −ul in the subspace of vectors that satisfy the constraint equation Bl ul = 0 is analysed. The pressure error does not play a role. In the present paper, due to the parameters contained in the left-upper Al -block we are interested in the dependence of the multigrid convergence behaviour on the scaling of the Al -block. This dependence can be analysed very easily (Lemma 3) if we present the analysis from [7] in a different form in which both the errors in the velocity and pressure components are taken into account. In this modified analysis we consider the smoother as in (4) and not the one (as in [7]) in which Dl is replaced by the identity. The analysis is formulated in terms of a smoothing and approximation property. We use the following norms. By · we denote the Euclidean norm on Rk . On Rnl +m l we also use the following norm: 2 2 ul u I 0 l n l (9) with l := := ul 2 +h l2 pl 2 = l pl 0 h l Im l pl h Corresponding matrix norms are denoted by ·, ·h also. For the Braess–Sarazin smoother we have the following result. Lemma 1 For method (4), with iteration matrix Sl , the following holds: In l 0 1 Al Sl = Al Sl1 0 0 Al Sl1 h = Al Sl1

Dl e(1 −2)+1

(10)

if max (Dl−1 Al ), 1 2

Proof −1/2 −1/2 Al Dl , The result in (10) follows from Bl ul = 0 and (6). Introduce A˜ l = Dl A simple computation shows that for the iteration matrix we have 1/2 −1/2 (Inl − B˜ lT ( B˜ l B˜ lT )−1 B˜ l )(Inl −−1 A˜ l ) 0 0 Dl Dl Sl = 0 Im l 0 0 ( B˜ l B˜ lT )−1 B˜ l (Inl − A˜ l )

(11)

−1/2 B˜ l = Bl Dl .

0 0

The operator Tl := Inl − B˜ lT ( B˜ l B˜ lT )−1 B˜ l is an orthogonal projector on Kern( B˜ l ), thus B˜ l Tl = 0. With Ml := Tl (Inl −−1 A˜ l )Tl , we obtain, for 1 2, −1/2 Ml1 −1 0 0 Dl 1 1 −1 Sl = Sl Sl = 0 Im l ( B˜ l B˜ lT )−1 B˜ l (Inl −−1 A˜ l )Ml1 −2 0 1/2 0 Tl (Inl −−1 A˜ l )Dl (12) × 0 0 Copyright q



18


Note that

−1/2

Dl

0

0

Al

Im l

−1/2

0

Dl

0

=

Im l

A˜ l

B˜ lT

B˜ l

0

Combining this with (12) yields

−1/2

0

Dl

0

Im l

Al Sl1 =

A˜ l Ml1 −1 +(Inl − Tl )(Inl −−1 A˜ l )Tl Ml1 −2 0

×

Tl (Inl −−1 A˜ l )Dl

1/2

0

0 0

0

(13)

0

From this and (10) it follows that l Al Sl1 l−1 = Al Sl1 and thus the identity in (11) holds. From (13) we obtain for max (Dl−1 Al ) = max ( A˜ l ) and 1 2, −1/2 D

1/2 Al Sl1 Dl 1/2

Dl

l

0

0 Im l

Al Sl1

A˜ l Ml1 −1 +(Inl − Tl )(Inl −−1 A˜ l )Tl Ml1 −2 Tl Inl −−1 A˜ l Dl

1/2

A˜ l Ml1 −1 +(Inl − Tl )(Inl −−1 A˜ l )Tl Ml1 −2 Dl

(14)

Note that Ml is symmetric positive definite with (Ml ) ⊂ [0, 1]. Using this and Tl A˜ l Tl = (Tl − Ml ) we obtain A˜ l Ml1 −1 +(Inl − Tl )(Inl −−1 A˜ l )Tl Ml1 −2 = ( A˜ l Ml − A˜ l Tl +(Tl − Ml ))Ml1 −2 = ( A˜ l −Inl )(Ml − Tl )Ml1 −2 A˜ l −Inl (Ml − Tl )Ml1 −2 = A˜ l −Inl (Ml − Inl )Ml1 −2 Using this in (14) we obtain the inequality in (11).

e(1 −2)+1

This analysis of the smoothing property applies only if problem (7) is solved exactly. In [10] a similar smoothing property is shown for the case that problem (7) is solved approximately. We further comment on this in Remark 3. We now consider the approximation property. Copyright q



19


Lemma 2 Take = 0 and = 1 in (2). Assume that is such that problem (2) is H 2 -regular. Then there exists a constant C A independent of l such that In l 0 −1 −1 (15) (Al − Pl Al−1 Rl ) C A Dl −1 0 0 h

holds. Proof ˜ }1i m be the nodal bases in Vl and Q l and Let {i }1i nl , { i l Fl u :=

nl

u i i

and

F˜l p :=

i=1

ml

˜ pi i

i=1

the finite element isomorphisms Rnl → Vl and Rm l → Q l , respectively. On Rnl we use a scaled nl 3 Euclidean inner product v, ul = h l i=1 vi u i , and similarly on Q l . The norms ul (pl ) and Fl u L 2 ( F˜l p L 2 ) are uniformly (w.r.t. u, p and l) equivalent. Let Al (and Al−1 ) be scaled such that

Al u, vl = (∇ Fl u, ∇ Fl v),

Bl u, pl = (div Fl u, F˜l p)

for all u, v ∈ Rnl , p ∈ Rm l

(16)

For fl ∈ Vl let u ∈ H01 ()3 , p ∈ L 20 () be the solution of (∇u, ∇v)−(div v, p) = (fl , v) for all v ∈ H01 ()3 (div u, q) = 0

(17)

for all q ∈ L 20 ()

Let (ul , pl ) be the Galerkin solution of this problem in the pair of spaces (Vl , Q l ). The matrix Al is the matrix representation of the finite element discretization of problem (17), cf. (16). Using this, the approximation property of the spaces (Vl , Q l ) and standard finite element techniques (duality argument) we obtain In l 0 u−ul L 2 +h l p − pl L 2 −1 −1 ch ˜ l2 (18) (Al − Pl Al−1 Rl ) c sup fl L 2 0 0 fl ∈Vl h

Using the scaling of Al as in (16) and standard properties of the finite element nodal basis we obtain Dl ch l−2 with a constant c>0. Thus, we obtain the bound in (15). I

In the approximation property it is important to have the projection factor ( 0nl 00 ) in (15). Without this factor one has to consider the Stokes problem in (17), where in the second equation the right-hand side 0 is replaced by (gl , q) with a gl ∈ Q l . The regularity properties of such a problem are in general less favourable as for the case with a 0 right-hand side. In particular, for H 2 -regularity one has to assume certain compatibility conditions on gl that are not satisfied for all gl ∈ Q l , cf. [11]. We further comment on this in Remark 3. Combination of the smoothing and approximation property yields a two-grid convergence result. Copyright q



20


Theorem 1 Take = 0 and = 1 in (2). Assume that is such that problem (2) is H 2 -regular. For the iteration matrix of the two-grid method with 2 = 0 the following holds: Ml h

CA e(1 −2)+1

for 1 2

with a constant C A independent of l. Proof Due to (10) we have −1 Ml = (Al−1 − Pl Al−1 Rl )

In l

0

0

0

Al Sl1

The desired result follows from (11) and Lemma 2.

In [7] such a result is proved for a two-grid method in which an additional projection step is used in the coarse-grid correction. In [12], however, it is noted that this projection step is superfluous. Using this two-grid contraction number bound one can derive a multigrid W -cycle convergence result using techniques from [13, 14]. The numerical experiments in Section 6 clearly show that the multigrid W -cycle method with Braess–Sarazin smoother is robust with respect to variation in the problem parameters and . We do not know of any analysis in which such a robustness property is proved. An elementary scaling argument can be used to derive a robustness result for the (less interesting) case = 0, >0 arbitrary. For this scaling argument we introduce the notation Il, =

In l

0

0

Im l

,

I˜l, =

Inl

0

0

Im l

(19)

with Ik the identity matrix in Rk . For = 0 let Ml, be the iteration matrix of the two-grid method with 1 pre- and 2 post-smoothing iterations (4) applied to the matrix Al, :=

Al

BlT

Bl

0

(20)

Note that Ml,1 = Ml as in (8). The effect of the scaling on the two-grid iteration matrix is given in the following lemma. Lemma 3 For = 0 the relation Ml, = Il, Ml,1 Il,−1 holds. Copyright q




21

Proof Let Sl, be the iteration matrix of the Braess–Sarazin smoother applied to Al, . The following relations hold: Al, = I˜l, Al,1 Il,−1 ,

Sl, = Il, Sl,1 Il,−1 ,

Il,−1 Pl Il−1, = Pl ,

−1 ˜ I˜l−1, Rl Il, = Rl

(21)

Using these we obtain 2 1 −1 −1 Ml, = Sl, (Il − Pl Al−1, Rl Al, )Sl, = Il, Ml,1 Il,

and thus the result is proved. Corollary 1 Introduce the norm

2 2 ul ul h2 −1 := l Il, = ul 2 + 2l pl 2 pl pl h,

with a corresponding matrix norm denoted by ·h, . Then we have Ml, h, = Ml,1 h = Ml h and thus the convergence result in Theorem 1 immediately yields an analogous result for the multigrid method applied to the scaled system. The Braess–Sarazin smoother has also been analysed in the setting of a cascadic multigrid method for Stokes problems in [8, 9]. In this analysis, it is assumed that problem (7) is solved exactly. Remark 3 The analysis presented above applies to the case in which problem (7) is solved exactly. In [10], the smoothing property of the Braess–Sarazin method (and other methods of inexact Uzawa type) with an inexact solve in (7) is analysed. The results from that paper can be combined with a suitable approximation property as derived in [15, 16] that differs slightly from the one in (18). We outline the main results of this analysis and present the approximation and smoothing properties in our notation. The analysis in [16] can be used to derive the following approximation property (with l the scaling matrix from (9)): −1 −1 Rl )l2 h = l (Al−1 − Pl Al−1 Rl )l ch l2 (Al−1 − Pl Al−1

(22)

A proof of this result is given in the Appendix. We introduce the scaled matrix h l−1 BlT Al −1 −1 ˜ l := Al = A l l h l−1 Bl 0 ˜ l instead of Al . For most smoothers, ˜ l be the iteration matrix of the smoother applied to A Let S ˜ l = l Sl −1 . We assume that the in particular those of Braess–Sarazin type, cf. [10], we have S l latter relation holds. For the two-grid iteration matrix we obtain −1 Ml h = l (Al−1 − Pl Al−1 Rl )l l−1 Al Sl1 l−1

1

1

−1 ˜ l ch 2 A ˜l ˜ lS ˜ lS Rl )l A l (Al−1 − Pl Al−1 l

Copyright q



22


˜ l is the same as in [10] (cf. (5) in that paper) and results from [10] The scaling in the matrix A can be used to derive smoothing properties. For example, from Lemma 2 and Theorem 2 in [10] we obtain ˜ lS ˜ l1 c h −2 , 1 2 A 1 −1 l for a Braess–Sarazin smoother with an inexact (but sufficiently accurate, cf. [10]) inner solve in (7). This leads to a two-grid convergence result as in Theorem 1 also for the case of an ‘inexact’ Braess– Sarazin smoother. Note, however, that such a result can be shown only for problem parameters = 0, = 1 (or some other fixed positive constant). Remark 4 In [17], a modification of the Braess–Sarazin smoother for the case in which problem (7) is solved (0) approximately is suggested. The starting velocity ul is usually not in or close to the kernel of (0) Bl , i.e. Bl ul can be ‘large’. This may have a significant (negative) influence on the pressure correction term pˆ l in (7), in particular if the parameter is chosen relatively large (cf. numerical experiments in Section 6). To avoid this, the following modification is suggested in [17]. Suppose that a sequence of pre- or post-smoothing iterations have to be performed. After the first of these (0) (1) iterations only the velocity update ul → ul is computed but for the pressure one keeps the old (1) (0) value, pl := pl . In the subsequent −1 iterations the standard method without this modification is applied. 3.2. Vanka smoother The Vanka-type smoothers, originally proposed by Vanka [18] for finite difference schemes, are block Gauß–Seidel type of methods. If one uses such a method in a finite element setting then a block of unknowns consists of all degrees of freedom that correspond with one element. Numerical tests given in [5] show that the use of this element-wise Vanka smoother can be problematic for continuous pressure approximations. In [5], the pressure-oriented Vanka smoother for continuous pressure approximations has been suggested as a good alternative. In this method a local problem corresponds to the block of unknowns consisting of one pressure unknown and all velocity degrees of freedom that are connected with this pressure unknown. We only consider this type of Vanka smoother. We first give a more precise description of this method. We take a fixed level l in the discretization hierarchy. To simplify the presentation we drop the level index l from the notation, i.e. we write, for example, ( up ) ∈ Rn+m instead of ( upll ) ∈ Rnl +m l . ( j)

Let r P : Rm → R be the pressure projection (injection) ( j)

rP p= p j ,

j = 1, . . . , m

For each j (1 jm) let the set of velocity indices that are ‘connected’ to j be given by ( j)

V j = {1in | (r P B)i = 0} Define d j := |V j | and write V j = {i 1 0, A and S be spectral bounds such that A Q A A A Q A

(26)

S Q S S S Q S

(27)

and

Below we first specify the choice of Q A and of Q S for the discrete generalized Stokes problem. Then we discuss the PMINRES and inexact Uzawa method. 4.1. Preconditioners for Q A and Q S The matrix A has block-diagonal form with identical blocks. Such a block corresponds to the finite element discretization of a scalar reaction–diffusion problem of the form −u +u = f . For Q A we use one iteration of a symmetric V-cycle multigrid method (for each of the blocks in A). In [22] it is shown that for this preconditioner the inequalities A Q A AQ A

(28)

hold with a constant A >0 independent of l, and . Note that the upper spectral constant is A = 1. For typical multigrid methods, the spectral constant A is close to one (typically A 0.85). We now discuss the choice of Q S . For this we introduce an auxiliary Neumann problem in the pressure space, with a given g ∈ L 2 (): Find w ∈ H 1 ()∩ L 20 () such that (∇w, ∇) = (g, )

for all ∈ H 1 ()∩ L 20 ()

Let N = Nl be the stiffness matrix resulting from a finite element discretization of this problem in the pressure finite element space Q l . Let M = Ml be the mass matrix for the pressure space. Let Q N be a preconditioner of N induced by one symmetric V-cycle multigrid iteration applied to the discrete problem with stiffness matrix N . The (Cahouet–Chabard) Schur complement preconditioner Q S is given by −1 Q −1 +Q −1 S := M N ,

= max{, h l2 }

(29)

In [23, 24] it is shown that under certain regularity assumptions for the Stokes problem this preconditioner has corresponding spectral bounds S >0, S in (27) that are independent of the parameters l, and . 4.2. The PMINRES method In the PMINRES method used for solving a linear system with matrix A we use a block-diagonal preconditioner defined by QA 0 M= (30) 0 QS Copyright q



26


For a discussion and an efficient implementation of the PMINRES method we refer to the literature [25, 26]. In an efficient implementation one needs per PMINRES iteration one evaluation of Q −1 A , one evaluation of Q −1 and one matrix–vector product with A (and a few other inexpensive S operations). Let r(k) be the residual in the kth iteration of this method. The convergence of the PMINRES can be analysed based on the well-known residual bound r(k) M−1 min max | pk ()| r(0) M−1 pk ∈k , pk (0)=1 ∈(M−1 A)

(31)

For the spectrum of the preconditioned matrix M−1 A the following result is given in [27, 28]: (M−1 A) ⊂ 12 A − 2A +4 S A , 12 A − 2A +4 S A ∪

A , 12

2 A + A +4 S A

(32)

This general result implies that the rate of convergence of the PMINRES method is robust with respect to variation of parameters (in our case: l, and ) if the spectral constants in (26) and (27) do not depend on these parameters. For our choice of the preconditioners this is indeed the case and thus the PMINRES method with Q A and Q S as explained in Section 4.1 has a rate of convergence that is robust with respect to variation in the parameters l, and . Remark 5 Consider the special case of only a rescaling of the Al -block in the matrix Al as in (20). The multigrid preconditioner and the Cahouet–Chabard Schur complement preconditioner automatically take this scaling into account. Let M be the block preconditioner as in (30) for A = Al, . −1 −1 −1 −1 Then, M−1 A = I M1 A1 I (with I = Il, as in (19)) and thus (M A ) = (M1 A1 ) for all >0. 4.3. The inexact Uzawa method For the derivation of the inexact Uzawa method we consider the exact block factorization of the matrix A A 0 I A−1 B T A= (33) B −I 0 S An approximate Schur complement is given by T Sˆ = B Q −1 A B

(34)

−1 ≈ Sˆ −1 we obtain the Using the block factorization (33) and substituting A−1 ≈ Q −1 A and S approximate inverse of A −1 ˆ −1 −1 Q I −Q B S 0 A A A−1 ≈ (35) −B Q −1 −I 0 Sˆ −1 A

Copyright q




27

In general, the application of Q A is feasible, but Sˆ −1 w cannot be determined with acceptable computational costs. Therefore, we use Sˆ −1 w ≈ (w) with (w) the result of a PCG method with ˆ = w such that zero starting vector and preconditioner Q S applied to Sz (w)−z Sˆ z Sˆ

(36)

holds for some prescribed tolerance 0 s.t. a0 (x)c0

a.e. in

(26)

Let 1 = {(x1 , 0) : 0x1 1}, 2 = {(0, x2 ) : 0x2 1}, 3 = {(x1 , 1) : 0x1 1}, and 4 = {(1, x2 ) : 0x2 1} be the four sides of and * = 1 ∪2 ∪3 ∪4 . Let be the usual trace operator, i.e. : v ∈ H 1 () −→ v ∈ L 2 (*) be a continuous linear mapping such that v = v|* . Note that since v is continuous, its restriction to * is a continuous function on *. Thus, if ⊂ R2 , the graph of v|* can be drawn as a continuous curve above *. We choose V = {v ∈ H 1 () : v|1 = v|3 , v|2 = v|4 }

(27)

as the space of test functions. Then the corresponding variational or weak formulation to the problem (18)–(22) is Find (u, ) ∈ V ×R such that a(u, v) = (u, v) L 2 () ∀v ∈ V

(28)

where a(u, v) =

*u *v ai j +a0 uv dx *x j *xi i, j=1 2

and

(u, v) L 2 () =

uv dx

(29)

Theorem 2.1 Let (u, ) ∈ C 2 ()×R solve (18) with boundary conditions (19)–(22). Then (u, ) is a solution to the variational problem (28). Copyright q



60

C.-S. CHIEN, B.-W. JENG AND Z.-C. LI

Proof Let (u, ) ∈ C 2 ()×R satisfy (18) with boundary conditions (19)–(22) and let v ∈ V . Then Green’s formula yields (u, v) L 2 () = uv dx = uv dx

=

2

* − i, j=1 *x i

*u ai j +a0 u v dx *x j

2

*u *v = ai j dx − *x j *xi i, j=1 =

*u v ds + * *n a

a0 uv dx

*u *v ai j +a0 uv dx *x j *xi i, j=1 2

= a(u, v)

Note that the boundary term ‘ * (*u/*n a )v ds’ vanishes for v ∈ V because v |1 = v |3 , v |2 = v |4 , *u/*n a |1 = −*u/*n a |3 , and *u/*n a |2 = −*u/*n a |4 . Thus, (u, ) is also the solution to the variational problem (28). Next, by following the arguments in Hall and Porsching [13] we will prove the companion result, namely, that a solution to the variational problem (28) solves the classical eigenvalue problem (18)–(22). Theorem 2.2 Let (u, ) ∈ V ×R solve the variational problem (28) and suppose that u ∈ C 2 () and *u/*n ∈ C(*). Then (u, ) solves the eigenvalue problem (18) with boundary conditions (19)–(22). Proof Suppose that (u, ) satisfies the variational problem (28) and u ∈ H 1 (). From (28) and Green’s formula, we have 2 2 * *u *u *v *u ai j +a0 u v dx = − ai j dx − v ds + a0 uv dx *x j *x j *xi i, j=1 * *n a i, j=1 *x i

*u v ds * *n a *u = (u, v) L 2 () − v ds * *n a *u = uv dx − v ds ∀v ∈ V * *n a

= a(u, v)−

Copyright q



¨ COMPUTATION OF WAVE FUNCTIONS OF THE SCHRODINGER–POISSON SYSTEM

61

Thus, we obtain the following equation:

2

* − i, j=1 *x i

*u *u v ds = 0 ai j +a0 u −u v dx + *x j * *n a

∀v ∈ V

(30)

Since v is arbitrary, we obtain

2

* − i, j=1 *x i

*u ai j +a0 u −u v dx = 0 *x j

∀v ∈ C 1 ()

(31)

and

*u v ds = 0 ∀v ∈ V * *n a

(32)

Also since v is arbitrary in C 1 (), the integrand of (31) must be zero, i.e. 2

* − i, j=1 *x i

*u ai j +a0 u = u *x j

in

(33)

Below, let us prove (33). The cutoff function is defined as

(x) =

e1/(|x| 0

2 −1)

for |x|0 in the neighborhood of (x0 , y0 ): R (x0 , y0 ) = {(x, y)|(x − x0 )2 +(y − y0 )2 <} Copyright q



62


due to the property of continuous functions. We design a specific function v=

e1/((

x−x0 2 y−y0 2 ) +( ) −1)

, x ∈ R (x0 , y0 )

0

otherwise

Obviously, v ∈ C 1 (S), then

g v ds

R (x0 ,y0 )

g0 v dsg0

R (x0 ,y0 )

v ds

g0 >0 where >0. This contradicts (31). Next, we obtain from (32) 1 ∪3

*u v ds = *n a

1

v(x1 , 0)

0

*u *u (x1 , 0)+ (x1 , 1) dx1 = 0 *n a *n a

∀v ∈ V, v|2 = v|4 = 0

1 *u *u *u v ds = v(0, x2 ) (0, x2 )+ (1, x2 ) dx2 = 0 *n a *n a 2 ∪4 *n a 0 ∀v ∈ V, v|1 = v|3 = 0

(34)

(35)

Also since v ∈ H 1/2 (*) (due to v ∈ H 1 ()) is arbitrary, and the cutoff function in one dimension, v ∈ H 1/2 (*), based on the above arguments, we obtain from (34) and (35) *u *u (x1 , 0) = − (x1 , 1), *n a *n a

0x1 1

*u *u (0, x2 ) = − (1, x2 ), *n a *n a

0x2 1

which are the boundary conditions (20) and (22). The boundary conditions (19) and (21) follow since u ∈ V . Therefore, (u, ) satisfies (18) with boundary conditions (19)–(22). From Theorems 2.1 and 2.2 we have that the eigenvalue problem (18)–(22) and the variational problem (28) are basically equivalent. 2.2. Existence of eigenpairs of the variational problem We will show that the bilinear form a(·, ·) and the space V defined in (29) and (27), respectively, satisfy some important properties. In what follows the notations · m, and |·|m, denote the Sobolev norm and seminorm in the mth order Sobolev space H m () = W m,2 (), m ∈ N∪{0}, respectively. Note that H 0 () = L 2 () and u 0, = |u|0, = u L 2 () . Copyright q




63

Proposition 2.3 The assumptions (23)–(26) imply that the bilinear form a(·, ·) defined in (29) is symmetric, bounded, and coercive on V × V . Proposition 2.4 The space V = {v ∈ H 1 () : v|1 = v|3 , v|2 = v|4 } defined in (27) is dense and compactly imbedded in L 2 (). Moreover, it forms a closed subspace of H 1 (). From Propositions 2.3 and 2.4 we may consider the variational problem (28) in the general framework of abstract elliptic EVPs in Hilbert spaces. In order to study the existence of eigenpairs of the variational problem (28), it is useful to introduce the linear operator T : L 2 () −→ V defined by T f ∈ V,

a(T f, v) = ( f, v) L 2 ()

∀v ∈ V

(36)

See References [15, p. 671; 16, Section 6.2]. Then T is the solution operator for the boundary value problem: 2

* L(u) := − i, j=1 *x i

*u ai j +a0 u = f *x j

in

(37)

with periodic boundary conditions (19)–(22), i.e. u = T f solves (37) with periodic boundary conditions (19)–(22). Thus, T is the inverse of the differential operator L, considered on functions that satisfy the boundary conditions (19)–(22). Note that if f ∈ L 2 (), then the linear form v −→ (v) = ( f, v) L 2 () is bounded (or continuous) on V , i.e. ∃c>0

s.t. |(v)|c v 1,

∀v ∈ V

Since the bilinear form a(·, ·) : V × V −→ R is symmetric, bounded, and coercive on V × V and the linear functional (·) : V −→ R is bounded on V , the Lax–Milgram theorem shows that Equation (36) has a unique solution T f for each f ∈ L 2 (). Moreover, since T f 21, a(T f, T f ) = ( f, T f ) L 2 () f L 2 () T f L 2 () f L 2 () T f 1, we have that 1

T f 1, f L 2 ()

∀ f ∈ L 2 ()

(38)

where >0 is the coercivity constant of a(·, ·). Further, the linear operator T has the following properties. Proposition 2.5 The solution operator T defined on V is compact, i.e. T (B) is precompact in V for every bounded subset B of V . Copyright q



64


Proposition 2.6 T : V −→ V is a self-adjoint and positive operator on V with the inner product a(·, ·), that is, (i) self-adjoint: a(T u, v) = a(u, T v) for all u, v ∈ V , (ii) positive: a(T v, v)>0 for all 0 = v ∈ V . Now, the existence of exact eigenpairs for the variational problem (28) can be stated and proved as follows. Theorem 2.7 (i) The eigenvalues i (i = 1, 2, . . .) of the variational problem (28) form an increasing sequence 00 independent of h such that inf {|v −vh |0, +h|v −vh |1, }Ch s+1 v s+1,

vh ∈Vh

∀v ∈ V ∩ H s+1 (), 1sk

(41)

Proof Consider v ∈ V ∩ H s+1 (). Let h v ∈ X h be the usual piecewise Lagrange interpolant of v with respect to the FE-mesh, i.e. let h v =

r −1 c

v(a jc+i ) jc+i

j=0 i=1

Since v ∈ V satisfies the boundary conditions v|1 = v|3 and v|2 = v|4 , we have that v(ai ) = v(a(r −1)c+i ), v(a jc+1 ) = v(a( j+1)c ),

i = 2, . . . , c −1 j = 1, . . . ,r −2

v(a1 ) = v(ac ) = v(a(r −1)c+1 ) = v(ar c ) Then from (39) and the direct sum (40), we obtain h v =

r −2 c−1

v(a jc+i ) jc+i +

j=1 i=2

+

r −2 j=1

c−1 i=2

v(ai )[ i + (r −1)c+i ]

v(a jc+1 )[ jc+1 + ( j+1)c ]+v(a1 )[ 1 + c + (r −1)c+1 + r c ] ∈ Vh

By the classical interpolation error estimates that have been established in Reference [17, Section 3.1], we have that |v −h v|0, Ch s+1 |v|s+1,

and |v −h v|1, Ch s |v|s+1,

Thus, inf {|v −vh |0, +h|v −vh |1, } |v −h v|0, +h|v −h v|1,

vh ∈Vh

Ch s+1 |v|s+1, Ch s+1 v s+1,

Now, the finite element approximation of the variational problem (28) is as follows: Find (u h , h ) ∈ Vh ×R such that a(u h , vh ) = h (u h , vh ) L 2 () Copyright q


∀vh ∈ Vh

(42)



(r _2)(c _1)+1 (r _ 3)(c _1)+1

2(c _1)+1 (c _1)+1

• • • • • •

• • (r _1)(c _1) • • (r _2)(c _1)

• • 3(c _1) • • 2(c _1) • • c _2 c _1

• • • • • • •1 • • 2

67

3

Figure 2. Renumbering of the nodes not belonging to 3 or 4 .

where a(·, ·) is given by (29). For this variational approximation problem (42), a discrete analogue of Theorem 2.7 holds. Theorem 3.3 Problem (42) has a finite sequence of eigenvalues 00 s.t. f (x)c0 a.e. in

Actually, (47) is also a special case of the well-known Kohn–Sham N -eigenfunction equation in condensed matter and quantum-chemistry computation: −n (r )+ f (r )n (r ) = n n (r ),

r ∈ R3 , n = 1 : N

(49)

where 2N is the number of electrons in the system, or their number per period in the case that f (r ) is a periodic function as we consider here. Note that N can be very large, see e.g. References [1, 2] and references cited therein. The variational formulation of problem (47) with boundary conditions (48) is Find (u, ) ∈ V ×R such that as (u, v) = (u, v) L 2 ()

∀v ∈ V

where the space V is defined by (27) and as (u, v) = (∇u ·∇v + f uv) dx

(50)

(51)

The corresponding finite element approximation to the variational problem (50) is Find (u h , h ) ∈ Vh ×R such that as (u h , vh ) = h (u h , vh ) L 2 () Copyright q


∀vh ∈ Vh

(52)


70


where the finite dimensional subspace Vh of V is given by (40). Note that the model (47)–(48) is a special case of the eigenvalue problem (18)–(22). Therefore, the discussions in Sections 2 and 3 also hold for the SE problem (47). Let h, h ∈ (0, 1) be chosen so that h = O( h 2 ). We divide the domain into two uniform triangulations T and T , and let V , V ⊂ V be the corresponding finite-dimensional subspaces, which h h h h are defined by (40). From the discussion in the above section, the finite element solutions on the coarse and fine grids to the SE problem (47) with (48) can be reduced to solve the following two generalized eigenvalue problems:

N h Find (c h , h ) ∈ R ×R such that K h c h = h M h c h

Find (ch , h ) ∈ R Nh ×R such that K h ch = h Mh ch

(53) (54)

× N N h h , and K h , Mh ∈ R Nh × Nh . In order to derive the where N h = dim V h , Nh = dim Vh , K h , M h ∈R two-grid scheme, we treat the generalized eigenvalue problems (53) and (54) as two nonlinear systems of equations:

F h (x, ) = (K h −M h )x = 0

(55)

Fh (x, ) = (K h −Mh )x = 0

(56)

and

N N h h and Fh : R Nh ×R → R Nh . We will discuss how the extremum eigenpairs where F h : R ×R → R h be the on the fine grid can be approximated by their counterparts on the coarse grid. Let I h

interpolation operator from the coarse space R Nh to the fine space R Nh . We have the following result. The proof is a slight modification given in Reference [9] and is omitted here. Theorem 4.1 Assume that (c h , h ) satisfies F h (c h , h ) = 0 and (ch , h ) is the zero point of Fh (x, ) = 0 on the fine grid. Then the approximate eigenvector c¯h on the fine grid is obtained by solving h K h c¯h = ch ) h Mh (I h

(57)

h : R N h → R Nh is the interpolation operator defined above. where I h

Note that Davidson’s method [26] has been reported to be quite successful for computing the lowest eigenvalues and their corresponding eigenvectors of symmetric eigenvalue problems in quantum chemistry. Davidson’s method is regarded as an extension of the Lanczos method, and from the viewpoint of numerical implementation it is more related to Arnoldi’s method [27]. In Reference [28], Sleijpen and van der Vorst combined Jacobi’s method with Davidson’s method called the Jacobi–Davidson method, which can efficiently improve the convergence rate of Davidson’s method and can be modified to compute nonextremal eigenvalues. Moreover, it can also be applied to solve generalized eigenproblems and polynomial eigenproblems [29]. However, a detailed implementation of the Jacobi–Davidson method for solving the SP system is beyond the scope of this paper and will be given elsewhere. Copyright q




71

Suppose that we already obtained the eigenpair on the coarse grid. Then the fine grid approximate eigenvector c¯h can be obtained by solving the linear system (57). To obtain the fine grid approximate eigenvalue ¯ h , we compute the Rayleigh quotient: c¯T K h c¯h ¯ h = Th c¯h Mh c¯h Actually the approximate eigenpair (c¯h , ¯ h ) is still not accurate enough. We use (c¯h , ¯ h ) as the initial point and perform the RQI to improve the accuracy of the eigenpair. Since the fine grid solution (c¯h , ¯ h ) serves only as starting initial guess for the RQI, it is unnecessary to compute the approximate eigenvector c¯h on the fine grid to the desired accuracy for solving the linear system (57). Thus, we propose to use the iterative method, e.g. CG method, MINRES, etc., to solve (57) only a few iterations. Suppose that (x1 , 1 ) is an eigenpair of the generalized eigenvalue problem K x = M x, and (x0 , 0 ) is an approximation to (x1 , 1 ). In the RQI, to obtain a better estimate of the eigenvector, we solve the following linear system: (K −0 M)z = M x0

(58)

However, this system is ill-conditioned as the approximate eigenvalue 0 is close to the exact one 1 . Instead of solving (58), we use the idea of Lui and Golub [30] to solve a rank-one modification of the coefficient matrix K −0 M, i.e. (K −0 M +(M x0 )(M x0 )T )y = M x0

(59)

Then we have (K −0 M)y = (1−(M x0 )T y)M x0 Thus, the solution of (58) is z = y/(1−(M x0 )T y). Our numerical results show that the technique can be used to handle both simple and multiple eigenvalues. Note that linear systems with multiple right-hand sides that appear in the corrector step of Algorithm 4.2 below can be effectively solved using the conjugate gradient method in Reference [31] or the MINRES in Reference [32]. The two-grid finite element discretization scheme for the SE problem (47) with boundary condition (48) is described as follows. Algorithm 4.2 A two-grid finite element discretization algorithm for computing the extremum eigenpairs of the SE problem (47) with periodic boundary conditions (48). Input := stopping criterion for the RQI. 1 m 1. Predictor step: Find the first m eigenvalues , . . . , and the corresponding eigenvectors h h

1 , . . . , cm ∈ R N h with [(ci )T M(ci )]1/2 = 1, i = 1, . . . , m, on the coarse grid such that c h h h h h i i i K = Mh c , h c h h h

Copyright q


i = 1, 2, . . . , m Numer. Linear Algebra Appl. 2008; 15:55–82 DOI: 10.1002/nla

72


2. Corrector step: (i) Perform a few iterations of the CG method or the MINRES for each SPD linear system on the fine grid: Find ch1 , . . . , chm ∈ R Nh such that i i K h chi = M (Ihh c ), h h h

i = 1, 2, . . . , m

(ii) Compute the Rayleigh quotient ih =

(chi )T K h (chi ) (chi )T Mh (chi )

,

i = 1, 2, . . . , m

(iii) Perform the RQI: For i = 1 : m do j = 0, x0i = chi /[(chi )T Mh (chi )]1/2 , i0 = ih While K h x ij −ij Mh x ij 2 > do Use MINRES to solve (K h −ij Mh +(Mh x ij )(Mh x ij )T )y ij+1 = Mh x ij for y ij+1 z ij+1 = y ij+1 /(1−(Mh x ij )T y ij+1 ) x ij+1 = z ij+1 /[(z ij+1 )T Mh (z ij+1 )]1/2 ij+1 = (x ij+1 )T K h (x ij+1 ) j = j +1 End While End For Note that Algorithm 4.2 can be easily modified to compute the extremum eigenvalues of the linear eigenvalue problem −u = u

in = (0, 1)2

(60)

with periodic boundary conditions (48). To study how many extremal eigenpairs on the coarse grid can be used as initial guesses for computing their counterparts on the fine grid, we quote the following two rules described in Reference [9] concerning the implementation of Algorithm 4.2. 1. Choose proper coarse and fine grids so that the multiplicity of eigenvalues on these two grids is the same. 2. Choose a proper number of eigenvalues to compute so that the multiplicity of eigenvalues on both grids is never missed. We remark here that there is a basic difference between two-grid discretization schemes and multilevel cycle iterations for solving elliptic eigenvalue problems. As we pointed out in Section 1, in the former the further correction on the coarse grid is replaced by the RQI because the second derivative of the operator equation with respect to the state variable is zero, see e.g. References [8, 9] for details. However, in the latter, we do not exploit the ideas of linear and quadratic approximation of the operator equations associated with elliptic eigenvalue problems. Therefore, it is necessary to perform coarse level updates of solutions combined with fine level iterations, which may increase the convergence rate by one order. We refer to References [1, 2, 33, 34] for details. Copyright q




73

At last but not the least, we wish to point out that for multiple eigenvalues, orthogonalization of the computed eigenvectors on the coarse grid is necessary, see e.g. References [33, 34]. For clustered eigenvalues as can be seen from our numerical examples in Section 5, without the orthogonalization process on the coarse grid, it might happen that the computed eigenvectors could mix together. A detailed investigation of the orthogonalization process in the two-grid discretization scheme combined with the Jacobi–Davidson method will be given elsewhere. 4.2. The SP system Suppose that we wish to compute the first k eigenpairs of the ESP, which is given in (11)–(14) with periodic boundary conditions on = (0, 1)2 . Since the ESP should be solved in an iterative way, we rewrite (11) as G(m , m , ) ≡ L(m , m )+ N (m ) = 0,

m = 1, 2, . . . , k, ∈ [0, 1]

(61)

where −m )m = 0 L(m , m ) = − 12 m +(W

(62)

is the linear SE problem, and N (m ) = W m

(63)

is the nonlinear term. Equation (61) is a parameter-dependent problem, which in general is solved by using numerical continuation methods. The ESP (11)–(14) is solved in Reference [9] using a two-grid centered difference continuation algorithm, where the nonlinear term 0 +1 |∇W |2 is set to be 1. In this section we will show how the algorithm described in Reference [9] can be generalized to solve the ESP, where the nonlinear Poisson equation is taken into account. For convenience we assume that the occurrence probability m of the initial states 0m (x) and the dopant density n ∗ are all constants, namely, m = c1 , m = 1, 2, . . . , k, and n ∗ = c2 . To begin with, we compute the first k eigenpairs of (61) with = 0 on the coarse grid. Next, we use the computed eigenvectors as the right-hand side for the nonlinear Poisson equation (12). That is, we solve k 2 2 −∇ ·(0 +1 |∇W | )∇W = c1 m −c2 (64) m=1

under the constraint (14) by an iterative way, where c1 and c2 are constants that satisfy the following equation: k 2 c1 (65) (n −n ∗ ) dx = m −c2 dx = 0

k

m=1

Equation (65) implies c1 m=1 2m dx = c2 dx. From the normalization condition (13), we obtain c1 k = c2 vol(), where vol() denotes the volume of the domain . Without loss of generality we choose c1 = 1, then c2 = k/vol(). Note that in the iterative process the nonlinear term |∇W |2 is obtained in the previous iteration that can be easily computed in the finite element code. Then we go back to compute the first k eigenpairs of (61) with = 0 , where the nonlinear potential W is the solution of (64) we just obtained. Afterward W is recomputed by solving (64) with Copyright q



74


constraint (12). We repeat the process described above by increasing the parameter from = 0 to 2 0 and so on until j 0 = 1 is reached. Algorithm 4.3 A continuation algorithm for solving the quasilinear ESP (11)–(14) on the coarse grid. Input c1 := occurrence probability of each initial state 0m (x). c2 := dopant density. 0 := stepsize for the continuation algorithm on the nonlinear term. 0 , 1 := constants. For = 0 : 0 : 1 do (i) (ii) (iii) (iv)

Compute the first k eigenpairs of (61). Compute the right hand side of (64). Solve the nonlinear Poisson equation (64) iteratively with the constraint (14). If = 1, go to (i) and update the eigenpairs until the eigenvalues converge.

End For We will use the eigenpairs obtained in Algorithm 4.3 as starting approximate eigenpairs for computing wave functions of the SP system. Algorithm 4.4 A two-grid finite element algorithm for computing wave functions of the SP system (1)–(4). 1 1 ), . . . , (k , k ) ∈ V ×R 1. Use Algorithm 4.3 to find the first k approximate eigenpairs ( , h h h h h and the approximate nonlinear potential W of the quasilinear ESP on the coarse grid. h 2.

(i) Solve the linear problems on the fine grid: Find 1h , . . . , kh ∈ Vh such that 1 2

∇m h ·∇v +

m m (W h + W )h v = h

m v h

∀v ∈ Vh , m = 1, 2, . . . , k

(ii) Compute 1h , . . . , kh by the Rayleigh quotient:

m m 1 m 2 h + W )(h ) 2 ∇h ·∇ h + (W m = , h m 2 (h )

m = 1, 2, . . . , k

(iii) Compute Wh iteratively by solving

k m 2 c1 (0 +1 |∇Wh | )∇Wh ·∇v = (h ) −c2 v 2

m=1

∀v ∈ Vh with

Wh = 0

where W h is used in the first iterate on the nonlinear term of the equation above. 3. (i) Update the eigenpairs (1h , 1h ), . . . , (kh , kh ) by performing the RQI on the fine grid. Copyright q



Copyright q

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0.00000000000 39.47970992989 39.47970992989 39.47970992989 39.47970992989 78.95941566262 78.95941566262 78.98971167237 78.98971167237 157.99425108728 157.99425108728 157.99425108728 157.99425108728 197.43661249913 197.43661249913 197.43661249913 197.43661249913 197.80106214674 197.80106214674 197.80106214674 197.80106214674

Coarse grid eigenvalue h 0.14[−13] 0.24[−12] 0.15[−12] 0.33[−12] 0.32[−12] 0.29[−11] 0.31[−11] 0.41[−12] 0.41[−12] 0.73[−12] 0.43[−12] 0.58[−12] 0.40[−12] 0.67[−13] 0.63[−13] 0.67[−13] 0.12[−12] 0.40[−12] 0.26[−12] 0.34[−12] 0.37[−12]

Residual 0.00000000000 39.47841808940 39.47841808940 39.47841808940 39.47841808940 78.95683594290 78.95683594290 78.95686187842 78.95686187842 157.91372922340 157.91372922340 157.91372922340 157.91372922340 197.39216784139 197.39216784139 197.39216784139 197.39216784139 197.39267588481 197.39267588481 197.39267588481 197.39267588481

Fine grid eigenvalue h 0.17[−12] 0.25[−3] 0.25[−3] 0.25[−3] 0.25[−3] 0.15[−3] 0.15[−3] 0.20[−2] 0.20[−2] 0.32[−2] 0.32[−2] 0.32[−2] 0.32[−2] 0.33[−2] 0.33[−2] 0.33[−2] 0.33[−2] 0.10[−1] 0.10[−1] 0.10[−1] 0.10[−1]

Residual — 39.47841792266 39.47841792266 39.47841792266 39.47841792266 78.95683584531 78.95683584531 78.95684348176 78.95684348176 157.91369078041 157.91369078043 157.91369078042 157.91369078042 197.39209915967 197.39209915967 197.39209915968 197.39209915968 197.39219455215 197.39219455216 197.39219455216 197.39219455216

RQI 1 — 0.41[−10] 0.41[−10] 0.41[−10] 0.41[−10] 0.44[−10] 0.45[−10] 0.38[−10] 0.38[−10] 0.32[−10] 0.45[−10] 0.32[−10] 0.32[−10] 0.42[−10] 0.42[−10] 0.40[−10] 0.40[−10] 0.51[−9] 0.51[−9] 0.51[−9] 0.51[−9]

Residual — — — — — — — — — — — — — — — — — 197.39219455217 197.39219455216 197.39219455216 197.39219455216

RQI 2 — — — — — — — — — — — — — — — — — 0.44[−10] 0.44[−10] 0.45[−10] 0.45[−10]

Residual

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2

Total RQIs

1 1 Table I. The first 21 eigenvalues of the discrete problem associated with (60) and (48), h = 32 , h = 256 .


202 = 197.3920880217872

162 = 157.9136704174297

82 = 78.9568352087149

42 = 39.4784176043574

02 = 0

Exact eigenvalues


75


76


Table II. Implementing Algorithm 4.2 for the second eigenpair of the linear eigenvalue problem (60) with periodic boundary conditions (48). h

|0,1 −4h | |0,1 −h |

h = h2

h

|0,1 − h |

1 16 1 64 1 256 1 1024

39.503929651960

0.255120[−1]

39.480038938238

0.162133[−2]

39.479389824073 39.478913854376

|0,1 −4h | |0,1 −h |

h

|0,1 − h |

39.498562771820

0.201452[−1]

15.73522

39.478498918695

0.813143[−4]

247.74435

0.972220[−3]

1.66766

39.478417922663

0.318305[−6]

255.46021

0.496250[−3]

1.95913

39.478417605636

0.127893[−8]

248.88461

(a) 1 4 1 8 1 16 1 32

0,1 = 42 = 39.47841760435743 h

h = h2

u 0,1 −u h L 2

1 16 1 64 1 256 1 1024

0.4039662[−2]

u 0,1 −u 4h L 2

u 0,1 −u h L 2

u 0,1 − x h L 2

u 0,1 −x 4h L 2

u 0,1 −x h L 2

(b) 1 4 1 8 1 16 1 32

0.1396944[−3]

0.2075774[−2]

1.94610

0.5643071[−6]

247.55030

0.6760986[−3]

3.07022

0.2208141[−8]

255.55756

0.2220860[−3]

3.04431

0.3966620[−9]

5.56681

(ii) Update Wh iteratively by solving the following nonlinear equation with constraint: k m 2 c1 (0 +1 |∇Wh |2 )∇Wh ·∇v = (h ) −c2 v ∀v ∈ Vh with Wh = 0

m=1

(iii) Repeat 3(i) and 3(ii) until it converge. 4. Compute wave functions of the SP system: −ih t m m h (x), h (x, t) = e m

m = 1, 2, . . . , k

5. NUMERICAL RESULTS We report some numerical results concerning the implementations of Algorithms 4.2 and 4.4. In 1 1 Examples 1–3, we choose h = 32 on the coarse grid and h = 256 on the fine grid, where the domain was divided into two uniform triangulations T and T of quadratic Lagrange triangles. The h h dimensions of the corresponding finite dimensional subspaces V h and Vh of V are 1024 and 65 536, respectively. The accuracy tolerance for solving linear systems of equations in the MINRES iterative method is 0.5×10−10 . Stopping criteria in Algorithm 4.2 for Examples 1 and 2 are 0.5×10−10 and 0.5×10−8 , respectively. All of our computations were executed on a Pentium 4 computer using FORTRAN 95 with double precision arithmetic. The notation [±n] stands for multiplication by 10±n . Copyright q



Copyright q


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

4.835701732025 44.357619875650 44.408221107723 44.410144158598 44.523835675750 83.945786462218 83.994302266952 84.064274267076 84.112790127502 162.884358737914 162.884359639090 162.951463469771 162.951468031607 202.459382770608 202.459383397733 202.469314414232 202.469314677017 202.827215552148 202.827216173706 202.850699598214 202.850699866593

Coarse grid eigenvalue h

0.56[−10] 0.53[−10] 0.11[−9] 0.30[−10] 0.42[−10] 0.37[−10] 0.37[−10] 0.35[−10] 0.47[−10] 0.44[−10] 0.36[−10] 0.34[−10] 0.45[−10] 0.42[−10] 0.38[−10] 0.32[−10] 0.47[−10] 0.40[−10] 0.35[−10] 0.42[−10] 0.28[−10]

Residual 4.835696282636 44.356275498174 44.406878428380 44.408818626900 44.522513943523 83.930773563951 83.983122143314 84.039999246285 84.092347876832 162.803418924235 162.803419827092 162.870731883247 162.870736453561 202.413165377865 202.413166622381 202.416539167221 202.416539672486 202.420638655748 202.420639894465 202.450822383808 202.450822897169

Fine grid eigenvalue h 0.20[−4] 0.35[−3] 0.35[−3] 0.33[−3] 0.33[−3] 0.14[−2] 0.12[−2] 0.18[−2] 0.16[−2] 0.36[−2] 0.36[−2] 0.35[−2] 0.35[−2] 0.35[−2] 0.35[−2] 0.39[−2] 0.39[−2] 0.10[−1] 0.10[−1] 0.10[−1] 0.10[−1]

Residual 4.835696281322 44.356275002153 44.406877940872 44.408818271399 44.522513578153 83.929400900088 83.980012584041 84.043087378481 84.093698937772 162.803368007865 162.803368910724 162.870684153830 162.870688724145 202.406702166668 202.406705557974 202.392170122472 202.392171109532 202.427515419411 202.427518742729 202.475192640022 202.475193596076

RQI 1 0.48[−10] 0.41[−10] 0.47[−10] 0.48[−10] 0.35[−10] 0.57[−6] 0.34[−5] 0.33[−5] 0.56[−6] 0.88[−10] 0.87[−10] 0.67[−10] 0.65[−10] 0.11[−3] 0.10[−3] 0.18[−3] 0.18[−3] 0.10[−3] 0.10[−3] 0.17[−3] 0.17[−3]

Residual — — — — — 83.929400802473 83.980003736173 84.043096098612 84.093699032324 — — — — 202.395210284989 202.395215772734 202.376995428612 202.376993732678 202.438916649295 202.438921788059 202.489818820045 202.489819720415

RQI 2 — — — — — 0.46[−10] 0.47[−9] 0.46[−9] 0.37[−10] — — — — 0.61[−4] 0.61[−4] 0.33[−4] 0.33[−4] 0.54[−4] 0.54[−4] 0.30[−4] 0.30[−4]

Residual — — — — — — — — — — — — — 202.391330613513 202.391335218160 202.376528066906 202.376528966700 202.441891424766 202.441896007628 202.490223282510 202.490224185363

RQI 3

— — — — — — — — — — — — — 0.55[−5] 0.55[−5] 0.14[−6] 0.14[−6] 0.36[−5] 0.36[−5] 0.11[−6] 0.11[−6]

Residual

— — — — — — — — — — — — — 202.391300939694 202.391305509998 202.376528050548 202.376528953401 202.441903919492 202.441908489799 202.490223287941 202.490224190792

RQI 4

— — — — — — — — — — — — — 0.32[−8] 0.32[−8] 0.50[−10] 0.48[−10] 0.89[−9] 0.89[−9] 0.50[−10] 0.50[−10]

Residual

1 1 1 1 1 2 2 2 2 1 1 1 1 4 4 4 4 4 4 4 4

Total RQIs

Table III. The first 21 eigenvalues and residuals of the corresponding eigenvectors of the discretized Schrödinger eigenvalue problem 1 1 h = 32 , h = 256 , implementing the MINRES with 20 ( f (x1 , x2 ) = 5+3 sin(2x1 )+2 cos(2x2 )) in 2D with periodic boundary conditions, iterations in Step 2(i) of Algorithm 4.2.


77



2

2

1.1

1

1

1 2

1 0.9

0.5 xis x -a

0.5 x -a 2 xis

(b)

1

1

u(x1,x2)

2

1 0

0.5 xis x -a

0 -1 -2 1

0.5 x -a 2 xis

(d)

1 0

0

0.5 xis x -a

0.5 x -a 2 xis

(e)

1

1 2

u(x ,x )

3 2 1 0 -1 -2 -3 1 0.5 x -a 2 xis

1 0 0

0

0

xis x 1-a

3 2 1 0 -1 -2 -3 1 0.5 x -a 2 xis

0.5 xis x 1-a

1

(h)

0 0

1

0 0

0.5 x -a 2 xis

0 0

(f )

1 0.5 xis x -a

0.5 xis x 1-a

1 0.5 xis x -a 1

3 2 1 0 -1 -2 -3 1

1 x -a0.5 2 xis

(i)

1

0.5 x -a 2 xis

3 2 1 0 -1 -2 -3 1

0.5

u(x1,x2)

-2 1

(c)

1

2

-1

0 0

0 -1 -2 1

1

u(x1,x2)

0 0

(a)

u(x1,x2)

-1

1 0.5 x -a 2 xis

u(x1,x2)

0 -2 1

0.8 1

(g)

u(x1,x2)

1.2

u(x ,x )

u(x1,x2)

78

0 0

0.5 xis x 1-a

Figure 3. The first nine eigenfunctions of the Schrödinger eigenvalue problem (47) with periodic boundary conditions (48), where the linear potential f (x1 , x2 ) = 5+3 sin(2x1 )+2 cos(2x2 ): (a) 1 = 4.835696281322; (b) 2 = 44.356275002153; (c) 3 = 44.406877940872; (d) 4 = 44.408818271399; (e) 5 = 44.522513578153; (f) 6 = 83.929400802473; (g) 7 = 83.980003736173; (h) 8 = 84.043096098612; and (i) 9 = 84.093699032324.

Example 1 The first test problem is the linear eigenvalue problem (60) with periodic boundary conditions (48). The exact eigenvalues of (60) with (48) are m,n = (2m)2 +(2n)2 where m, n are integers. We wish to find the first 21 eigenpairs of the discrete problem. Table I lists the numerical results, where we used the MINRES to solve each SPD linear system with 20 iterations in Step 2(i) of Algorithm 4.2. In order to study the convergence rate of Algorithm 4.2, 1 1 we compute the second eigenpair of (60) with (48) using h = h 2 with h = 14 , 18 , 16 , and 32 . Let h h (u 0,1 , 0,1 ) denote the second exact eigenpair of (60) with (48), and (u h , h ) and (x , ) denote the corresponding eigenpair on the fine grid and the RQI solution in Algorithm 4.2. From the Copyright q



Copyright q


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

4.836229977645 44.358013172448 44.408449803967 44.410370060726 44.523692294477 83.945873093816 83.994218699693 84.064006320436 84.112351987613 162.884711873319 162.884712767729 162.951601152006 162.951605679592 202.459315129779 202.459315750742 202.469236378615 202.469236638092 202.827125740886 202.827126355556 202.850513285949 202.850513551741

h

Coarse grid

0.12[−9] 0.12[−9] 0.13[−9] 0.11[−9] 0.13[−9] 0.12[−9] 0.12[−9] 0.12[−9] 0.14[−9] 0.12[−9] 0.13[−9] 0.12[−9] 0.13[−9] 0.12[−9] 0.11[−9] 0.11[−9] 0.11[−9] 0.12[−9] 0.12[−9] 0.12[−9] 0.13[−9]

Residual 4.836224545899 44.356668963592 44.407107288648 44.409044637633 44.522370657518 83.930869006993 83.983055975638 84.039713991958 84.091901018043 162.803773416990 162.803774313067 162.870870259607 162.870874795614 202.413108956794 202.413110189449 202.416511405152 202.416511903825 202.420537635642 202.420538860773 202.450585285435 202.450585793920

h in Step 2

0.19[−4] 0.35[−3] 0.35[−3] 0.33[−3] 0.33[−3] 0.14[−2] 0.12[−2] 0.18[−2] 0.16[−2] 0.36[−2] 0.36[−2] 0.35[−2] 0.35[−2] 0.35[−2] 0.35[−2] 0.39[−2] 0.39[−2] 0.10[−1] 0.10[−1] 0.10[−1] 0.10[−1]

Residual 4.836224778182 44.356668753808 44.407106796707 44.409044646935 44.522369959827 83.929492448116 83.979930497797 84.042817762015 84.093255817194 162.803722679059 162.803723575106 162.870822598807 162.870827134623 202.391304656417 202.391309192133 202.376580617464 202.376581513566 202.441742752436 202.441747288199 202.489905873022 202.489906769077

h in Step 3

Fine grid

0.49[−10] 0.47[−10] 0.49[−10] 0.34[−10] 0.50[−10] 0.49[−10] 0.49[−9] 0.47[−9] 0.49[−10] 0.80[−10] 0.84[−10] 0.67[−10] 0.64[−10] 0.40[−10] 0.42[−10] 0.51[−10] 0.50[−10] 0.38[−10] 0.45[−10] 0.50[−10] 0.50[−10]

Residual 1 1 1 1 1 2 2 2 2 1 1 1 1 5 5 4 4 5 5 4 4

RQIs 4.836227894180 44.356670961331 44.407108253891 44.409045712038 44.522369387239 83.929492604995 83.979929904671 84.042816281646 84.093253587692 162.803724767761 162.803725663788 162.870823417125 162.870827952874 202.391304566647 202.391309102290 202.376580655603 202.376581551691 202.441741912940 202.441746448637 202.489904274179 202.489905170220

h in Step 3

0.50[−10] 0.49[−10] 0.41[−10] 0.49[−10] 0.30[−10] 0.49[−10] 0.45[−10] 0.31[−10] 0.50[−10] 0.48[−10] 0.48[−10] 0.49[−10] 0.48[−10] 0.50[−10] 0.45[−10] 0.48[−10] 0.50[−10] 0.37[−10] 0.49[−10] 0.43[−10] 0.50[−10]

Residual 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

RQIs

4.836227889147 44.356670957600 44.407108251644 44.409045709966 44.522369388492 83.929492604250 83.979929905442 84.042816284244 84.093253591821 162.803724764366 162.803725660393 162.870823415792 162.870827951547 202.391304566645 202.391309102290 202.376580655200 202.376581551287 202.441741914458 202.441746450151 202.489904277123 202.489905173161

h in Step 3

Fine grid

0.18[−9] 0.17[−9] 0.20[−9] 0.15[−9] 0.21[−9] 0.14[−9] 0.17[−9] 0.20[−9] 0.23[−9] 0.18[−9] 0.18[−9] 0.18[−9] 0.18[−9] 0.17[−9] 0.17[−9] 0.16[−9] 0.16[−9] 0.20[−9] 0.20[−9] 0.21[−9] 0.21[−9]

Residual

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

RQIs

2 2 2 2 2 3 3 3 3 2 2 2 2 6 6 5 5 6 6 5 5

Total RQIs

Table IV. The first 21 eigenvalues of the discretized quasilinear ESP with periodic boundary conditions, computed by Algorithm 4.4 with 1 1 (x1 , x2 ) = 5+3 sin(2x1 )+2 cos(2x2 ) and 0 = 1, 1 = 0.3. h = 32 , h = 256 . The linear potential W


79


80


-0.7

0.65 Im(ψ )

1

1

Re(ψ )

0.6 0.55 0.5 0.45 1

-0.8 -0.9 1 1

1 0.5 x 2 axi s

0.5 x 2 axi s

0.5 0

0

xis x 1-a

Im(ψ ) 2

2

Re(ψ )

0.4 0.2

0 -1 1 0.5

x 2 axi s

1 0.5

0.5 0

0

0 -0.2 -0.4 1

1

x 2 axi s

s

xi x 1-a

0.04

2

0.02

1 6

Im(ψ )

6

0

1

1.5 1

Re(ψ )

0

0.5 xis x -a

0 -0.02

0.5 0

0

axis

x 1-

0 -1 -2 1

-0.04 1

1

1 0.5

x 2 axi s

0.5

0.5 0

0

x 2 axi s

s

xi x 1-a

0.5 0

0

s

xi x 1-a

Figure 4. The contours of the real and imaginary parts of the wave functions 1 , 2 , 6 of the SP system (x1 , x2 ) = 5+3 sin(2x1 )+2 cos(2x2 ), 0 = 1, and 1 = 0.3, at time t = 160. with W

numerical results shown in Table II, we have that |0,1 −h | ≈ O(h 4 ),

u 0,1 − x h L 2 ≈ O(h 4 )

Example 2 Consider the Schrödinger eigenvalue problem (47)–(48) with linear potential f (x 1 , x2 ) = 5+ 3 sin(2x1 )+2 cos(2x2 ). We implemented Algorithm 4.2 to find the first 21 eigenpairs of the discrete problem. Table III shows that the first 21 eigenvalues consist of one simple eigenvalue and four clusters of eigenvalues, where the first three clusters consist of four close eigenvalues, and the last cluster consists of eight close eigenvalues. The cluster structure is preserved on the fine grid. Figure 3 shows the first nine eigenfunctions of this problem. For the last eight eigenpairs we need to perform four RQIs. Copyright q




81

Example 3 Computing eigenpairs of the ESP and wave functions of the SP system. Consider the SP system (x1 , x2 ) = 5+3 sin(2x1 )+2 cos(2x2 ) and 0 = 1, 1 = 0.3. In order (1)–(4) with linear potential W to compare with the result of Example 2, we omitted the coefficient 12 in the Laplacian. We implemented Algorithm 4.4 to find the first 21 eigenpairs of the ESP and the corresponding wave functions of the SP system. Table IV lists the first 21 eigenvalues of the ESP and the residuals of the associated eigenvectors, which shows that the eigenvalues agree with those of the SE problem at least to two decimal digits. The contours of the real and imaginary parts of the wave functions 1 , 2 , and 6 at time t = 160 are displayed in Figure 4. 6. CONCLUSIONS We have presented a time-independent numerical scheme for computing wave functions of the SP system. By using the well-known separation of variables, the SP system is transformed into the ESP. A two-grid finite element discretization scheme is described to compute the first few eigenpairs of the ESP. The first few wave functions of the SP system can be easily obtained using (10). The proposed numerical scheme has the following advantages: 1. It is not necessary to discretize the left-hand side of (1), namely, the partial derivative of the wave function with respect to the time variable t. 2. The wave functions of the SP system at any time scale can be easily evaluated whenever the eigenpairs of the ESP are obtained. 3. The major computational difficulties such as closely clustered eigenvalues that occur in the SP system can be effectively computed using the proposed algorithm. 4. The rate of convergence of the proposed two-grid finite element discretization scheme is O(h 4 ), which is a significant improvement to the two-grid centered difference discretization scheme we proposed earlier. ACKNOWLEDGEMENTS

The authors would like to thank the two referees for their valuable comments that have improved the original version of this paper. Supported by the National Science Council of R.O.C. (Taiwan) through Project NSC 95-2115-M-005-004-MY3. REFERENCES 1. Brandt A, Bernholc J, Binder K (eds). Multiscale Computational Methods in Chemistry and Physics. NATO Science Series. IOS Press: Amsterdam, 2001. 2. Brandt A. Multiscale scientific computation: review 2000. Survery Report, Weizmann Institute of Science, Rehovot, Israel. 3. Illner R, Kavian O, Lange H. Stationary solutions of quasi-linear Schrödinger–Poisson systems. Journal of Differential Equations 1998; 145:1–16. 4. Ringhofer C, Soler J. Discrete Schrödinger–Poisson systems preserving energy and mass. Applied Mathematics Letters 2000; 13:27–32. 5. Chang S-L, Chien C-S, Jeng B-W. Computing wave functions of nonlinear Schrödinger equations: a timeindependent approach. Journal of Computational Physics 2007; 226:104–130. 6. Xu J. A novel two-grid method for semilinear elliptic equations. SIAM Journal on Scientific Computing 1994; 15:231–237. Copyright q



82


7. Chien C-S, Jeng B-W. A two-grid discretization scheme for semilinear elliptic eigenvalue problems. SIAM Journal on Scientific Computing 2006; 27:1287–1304. 8. Xu J, Zhou A. A two-grid discretization scheme for eigenvalue problems. Mathematics of Computation 1999; 70:17–25. 9. Chang S-L, Chien C-S, Jeng B-W. An efficient algorithm for the Schrödinger–Poisson eigenvalue problem. Journal of Computational and Applied Mathematics 2007; 205:509–532. 10. Costiner S, Ta’asan S. Simultaneous multigrid techniques for nonlinear eigenvalue problems: solutions of the nonlinear Schrödinger–Poisson eigenvalue problem in two and three dimensions. Physical Review E 1995; 52:1181–1192. 11. Schepper HD. Finite element methods for eigenvalue problems on a rectangle with (semi-)periodic boundary conditions on a pair of adjacent sides. Computing 2000; 64:191–206. 12. Schepper HD. A finite element method for differential eigenvalue problems with mixed classical and (semi-)periodic boundary conditions. Applied Mathematics and Computation 2001; 121:1–15. 13. Hall CA, Porsching TA. Numerical Analysis of Partial Differential Equations. Prentice-Hall: Englewood Cliffs, NJ, 1990. 14. Brenner SC, Scott LR. The Mathematical Theory of Finite Element Methods. Springer: New York, 1994. 15. Babuˇska I, Osborn J. Eigenvalue problems. In Finite Element Methods (Part 1), Ciarlet PG, Lions JL (eds), Handbook of Numerical Analysis, vol. II. North-Holland: Amsterdam, 1991; 641–787. ´ 16. Raviart PA, Thomas JM. Introduction a` l’Analyse Numérique des Equations aux Dérivées Partielles. Masson: Paris, 1983. 17. Ciarlet PG. The Finite Element Method for Elliptic Problems. North-Holland: Amsterdam, 1978. 18. Li ZC. Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities. Kluwer Academic Publishers: Dordrecht, 1998. 19. Babuˇska I. The finite element method with penalty. Mathematics of Computation 1973; 27:221–228. 20. Barrett JW, Elliott CM. Finite element approximation of the Dirichlet problem using the boundary penalty method. Numerische Mathematik 1986; 49:343–366. 21. Dussault JP. Numerical stability and efficiency of penalty algorithms. SIAM Journal on Numerical Analysis 1995; 32:296–317. 22. King JT. New error bounds for the penalty method and extrapolation. Numerische Mathematik 1974; 23:153–165. 23. King JT, Serbin SM. Computational experiments and techniques for the penalty method with extrapolation. Mathematics of Computation 1978; 32:111–126. 24. Shi ZC. On the convergence rate of the boundary penalty method. International Journal for Numerical Methods in Engineering 1984; 20:2027–2032. 25. Utku M, Carey GF. Boundary penalty techniques. Computer Methods in Applied Mechanics and Engineering 1982; 30:103–118. 26. Davidson ER. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real symmetric matrices. Journal of Computational Physics 1975; 17:87–94. 27. Saad Y. Numerical Methods for Large Eigenvalue Problems. Manchester University Press: Manchester, U.K., 1992. 28. Sleijpen GLG, van der Vorst HA. A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM Journal on Matrix Analysis and Applications 1996; 17(2):401–425. 29. Sleijpen GLG, Booten AGL, Fokkema DR, van der Vorest HA. Jacobi–Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 1996; 36(3):595–633. 30. Lui SH, Golub GH. Homotopy method for the numerical solution of the eigenvalue problem of self-adjoint partial differential operators. Numerical Algorithms 1995; 10:363–378. 31. Smith CF, Peter AF, Mittra R. A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields. IEEE Transactions on Antennas and Propagation 1989; 37:1490–1493. 32. Dul FA. MINRES and MINERR are better than SYMMLQ in eigenpair computations. SIAM Journal on Scientific Computing 1998; 19:1767–1782. 33. Beck TL. Multiscale methods for self-consistent electronic structure in real space. In Multiscale Computational Methods in Chemistry and Physics, Brandt A, Bernholc K, Binder K (eds), NATO Science Series. IOS Press: Amsterdam, 2001; 90–103. 34. Bernholc J, Briggs EL, Buongiorno Nardelli M, Fattebert JL, Ramamoorthy M, Schmidt WG, Sullivan DJ. Largescale multilevel solutions of Kohn–Sham equations: methodology and applications. In Multiscale Computational Methods in Chemistry and Physics, Brandt A, Bernholc J, Binder K (eds), NATO Science Series. IOS Press: Amsterdam, 2001; 65–89. Copyright q



NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2008; 15:83 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nla.573

Call for Papers: Special Issue on ‘Recent Advances in Computational Techniques for Biomedical Imaging’ Communications in Numerical Methods in Engineering (CNM) Communications in Numerical Methods in Engineering will publish a special issue on ‘Recent Advances in Computational Techniques for Biomedical Imaging’. The Guest Editors for this issue will be Professor Guo-Wei Wei, Department of Mathematics and Department of Electrical and Computer Engineering, Michigan State University and Professor Ge Wang, Director of Biomedical Imaging Division, Virginia Polytechnic Institute and State University. Recent advances in biomedical imaging have revolutionized diagnosis, therapy and discovery in both clinical and pre-clinical settings. Although traditional tomographic and post-processing methods become increasingly sophisticated, emerging new modalities play more and more critical roles in anatomical, functional, cellular and molecular imaging. Mathematical and computational techniques have been instrumental to these developments and their momentum. We invite the original papers and high-quality overviews on all the relevant topics including but not limited to: • • • • • •

Algorithms for biomedical image reconstruction Techniques for biomedical image analysis and visualization Forward models and inverse solutions Interface problems in biomedical imaging Multi-modal imaging Evaluation and comparison of biomedical imaging modalities

Paper Submission Details Completed papers should conform to the journal’s style and to the instructions for authors, which can be found on the journal’s homepage under ‘For Authors’. Papers have to be submitted online via the journal’s homepage to http://www3.interscience.wiley.com/cgibin/jabout/5902/ForAuthors.html under ‘Online Submission’. Authors must select Guo-Wei Wei or Ge Wang as Editor. Each submission must be clearly identified as a submission for this special issue, and the special issue code, RACTI, must be entered in the Special Issue Title field, when requested. Publication will take place according to the following schedule: Manuscript due: 1 April 2008 First review completion: Mid 2008 Anticipated publication date: End of 2008 Guest Editors Prof. Guo-Wei Wei Submitting Editor Department of Mathematics and Department of Electrical and Computer Engineering Michigan State University, MI, U.S.A. E-mail: [email protected] Prof. Ge Wang Director of Biomedical Imaging Division Virginia polytechnic Institute and State University, VA, U.S.A. E-mail: [email protected], [email protected]

Copyright q


Numerical Linear and Applications (No. 1, 2008)

Numerical Linear Algebra and Applications

Numerical Linear Algebra and Optimization, Vol. 1

Numerical linear algebra and optimization, Vol.1

Numerical Linear Algebra for Applications in Statistics

Numerical Methods and Applications

Numerical linear algebra

Numerical linear algebra

Numerical Linear Algebra

Applied numerical linear algebra

Numerical Linear Algebra

Applied numerical linear algebra

Numerical Linear Algebra

Numerical Linear Algebra

Numerical linear algebra

Applied Numerical Linear Algebra

Applied numerical linear algebra

Numerical Linear Algebra

Numerical linear approximation in C

Introduction to numerical linear algebra and optimisation

Linear and non linear numerical analysis of foundations

Numerical linear approximation in C

Numerical analysis 2000. Linear algebra

Linear Algebra and Its Applications

Linear Programming and Its Applications

Linear algebra and its applications

Linear Algebra and Its Applications

Numerical Methods in Computational Electrodynamics: Linear Systems in Practical Applications

Indefinite linear algebra and applications

Indefinite Linear Algebra and Applications

Linear Programming and its Applications

Numerical Linear and Applications (No. 1, 2008)

Numerical Linear Algebra and Applications

Numerical Linear Algebra and Optimization, Vol. 1

Numerical linear algebra and optimization, Vol.1

Numerical Linear Algebra for Applications in Statistics

Numerical Methods and Applications

Numerical linear algebra

Numerical linear algebra

Numerical Linear Algebra

Applied numerical linear algebra

Numerical Linear Algebra

Applied numerical linear algebra

Numerical Linear Algebra

Numerical Linear Algebra

Numerical linear algebra

Applied Numerical Linear Algebra

Applied numerical linear algebra

Numerical Linear Algebra

Numerical linear approximation in C

Introduction to numerical linear algebra and optimisation

Linear and non linear numerical analysis of foundations

Numerical linear approximation in C

Numerical analysis 2000. Linear algebra

Linear Algebra and Its Applications

Linear Programming and Its Applications

Linear algebra and its applications

Linear Algebra and Its Applications

Numerical Methods in Computational Electrodynamics: Linear Systems in Practical Applications

Indefinite linear algebra and applications

Indefinite Linear Algebra and Applications

Linear Programming and its Applications

Recommend Documents