Numer. Math. (2000) 86: 717–731 Digital Object Identifier (DOI) 10.1007/s002110000198
Numerische Mathematik
A cascadic multigrid algorithm for semilinear elliptic problems Gisela Timmermann Electrotechnical Institute, Technical University of Dresden, 01062 Dresden, Germany; e-mail:
[email protected] Received February 12, 1998 / Revised version received July 22, 1999 / c Springer-Verlag 2000 Published online June 8, 2000 –
Summary. We propose a cascadic multigrid algorithm for a semilinear elliptic problem. The nonlinear equations arising from linear finite element discretizations are solved by Newton’s method. Given an approximate solution on the coarsest grid on each finer grid we perform exactly one Newton step taking the approximate solution from the previous grid as initial guess. The Newton systems are solved iteratively by an appropriate smoothing method. We prove that the algorithm yields an approximate solution within the discretization error on the finest grid provided that the start approximation is sufficiently accurate and that the initial grid size is sufficiently small. Moreover, we show that the method has multigrid complexity. Mathematics Subject Classification (1991): 65N30, 65N55, 65F10
1. Introduction During the last years a special kind of “one-way multigrid” method was developed which uses a sequence of successively finer grids without projections onto coarser meshes. This type of cascadic multigrid algorithm was introduced by P. Deuflhard [Deu93] where he proposed a cascadic conjugate gradient method and demonstrated its high speed of convergence by numerical examples. In [Sha94a] V. V. Shaidurov proved optimality for this
Present address: Interactive Software GmbH, Basler Strasse 65, 79100 Freiburg, Germany; e-mail:
[email protected] Research of the author was supported by the Deutsche Forschungsgemeinschaft under grant GR 705/4-2
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algorithm in case of H 2 -regular second order elliptic equations. In this context optimality means accuracy, that is the iteration error on the finest grid is of the same order as the discretization error ul − u∗l ≈ u − ul where u is the exact solution of the continuous problem, ul denotes the solution of the discrete problem on the finest level, and u∗l is the corresponding approximation. Moreover, the algorithm has multigrid complexity, that is the complexity of the algorithm – denoted by comp(algo) – is proportional to the number of unknowns Nl on the finest grid comp(algo) = cNl . Later the optimality result was generalized in various respects: for twoand three-dimensional H 1+α -regular problems [Sha94b], [Bor94], [BD96]; for a larger class of iterative solvers [Bor94], [BD96]; for domains with curvilinear boundary [Sha95a]. In all papers cited only linear boundary value problems were considered. Recently, L. V. Gilyova and V. V. Shaidurov proposed a cascadic multigrid algorithm for a two-dimensional weakly nonlinear elliptic Dirichlet problem, see [GS98]. They used Newton’s method with frozen derivatives, i.e. with a fixed Jacobian evaluated at the coarsest grid, and the conjugate gradient method or a special Jacobi-type iteration for solving the arising linear problems. They proved optimality for this algorithm provided that the coarsest grid is sufficiently small and that the approximate solution on the coarsest grid is sufficiently accurate. In the present paper we show that this result still holds true if a full Newton method is used that is if the approximate solution from the previous grid is taken for linearization. Furthermore, we show that the Newton systems can be solved by more general basic iterative methods like traditional multigrid smoothers. 2. The cascadic multigrid algorithm We consider the boundary value problem
(1)
−
2
∂j (ajk ∂k u) + f (x, u) = 0 in Ω,
j,k=1
u = 0 on
∂Ω.
For the remainder of this paper we make the following assumptions. – Ω ⊂ R2 is a convex polygonal domain with boundary ∂Ω.
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– The functions ajk ∈ C 0,1 (Ω) are symmetric in the sense that ajk (x) = akj (x)
for all x ∈ Ω, j, k = 1, 2.
Moreover, they satisfy the uniform ellipticity condition (2)
2
a1 |ξ|2 ≤
ajk (x)ξj ξk ≤ a2 |ξ|2
j,k=1
¯ with some constants a1 , a2 > 0. for all ξ = (ξ1 , ξ2 )T ∈ R2 , x ∈ Ω – The nonlinear part f : Ω×R → R is a continuous function. Additionally, we suppose that the partial derivatives ∂2 f and ∂22 f exist and satisfy (3)
|∂2 f (x, v)| ≤ ω
and |∂22 f (x, v)| ≤ κ
for all x ∈ Ω, v ∈ R with some constants ω, κ > 0. These conditions imply that the boundary value problem (1) has a unique solution u ∈ H01 (Ω) ∩ H 2 (Ω), see [LU73], [Gri85]. The weak formulation reads: Find a function u ∈ H01 (Ω) which satisfies (4)
L(u, v) :=
2 Ω j,k=1
ajk ∂k u∂j v + f (x, u)v dx = 0
for all v ∈ H01 (Ω). This problem has a unique solution, too, namely the solution of (1). For approximating this solution we use linear finite elements. Let {Ti }li=0 be a nested sequence of triangulations which is constructed as follows: Starting with an initial triangulation T0 the triangulation Ti is obtained by connecting the midpoints of the edges of each triangle T ∈ Ti−1 . Then the mesh size parameters hi := maxT ∈Ti diam T satisfy hi =
hi−1 2
for i = 1, . . . , l.
The associated linear finite element spaces are defined by ¯ : v|T ∈ P1 (T ) for all T ∈ Ti , v|∂Ω = 0} Hi := {v ∈ C 0 (Ω) where P1 (T ) denotes the space of linear polynomials over the triangle T . Since the sequence of triangulations is nested we have H0 ⊂ H1 ⊂ · · · ⊂ Hl ⊂ H01 (Ω).
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The finite element approximation for problem (4) reads: Find a function ui ∈ Hi such that 2 L(ui , vi ) = ajk ∂j ui ∂k vi + f (x, ui )vi dx = 0 (5) Ω
j,k=1
for all vi ∈ Vi . Under the assumptions (2) and (3) this problem has a unique solution which satisfies the error estimates (6)
u − ui H 1 (Ω) ≤ chi uH 2 (Ω) ,
(7)
u − ui L2 (Ω) ≤ ch2i uH 2 (Ω)
where the constants c > 0 are independent of hi and u. At this point we remark that we will use the symbol c for any constant, in general. In particular, this means that it may have different values when used several times in an estimate. Those constants which are referred to later on are enumerated by c1 , c2 , etc. or denoted by special letters such as ω, κ, etc. For the solution of the nonlinear problems (5) we apply Newton’s method. Therefore, we define the bilinear form Lr : H01 (Ω) × H01 (Ω) → R with ¯ by r ∈ C 0 (Ω) 2 Lr (v, w) := (8) ajk ∂k v∂j w + ∂2 f (x, r)vw dx. Ω j,k=1
This bilinear form is the linearization of the form L(., .) at r. Due to (2) and (3) it is coercive, that is 2 Lr (v, v) = ajk ∂k v∂j v + ∂2 f (x, r)v 2 dx Ω j,k=1 (9) ≥ a1 |v|2H 1 (Ω) ≥ cv2H 1 (Ω) ¯ and the constant c > 0 is independent of r. for all v ∈ H01 (Ω), r ∈ C 0 (Ω), Thus, we can introduce the energy norm 1/2 (10) . [[v]]r := Lr (v, v) Together with Lr (v, v) ≤ |Lr (v, v)| ≤ a2 |v|2H 1 (Ω) + ωv2L2 (Ω) ≤ cv2H 1 (Ω) estimate (9) implies that this norm is uniformly equivalent to the H 1 -norm, i.e., there exists a constant c > 1 independent of r such that (11)
1 vH 1 (Ω) ≤ [[v]]r ≤ cvH 1 (Ω) c
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¯ for all v ∈ H01 (Ω), r ∈ C 0 (Ω). ∗ Let u0 ∈ H0 be an approximate solution of (5) on the coarsest grid i = 0. Then we fix i ∈ [1, l] and consider one Newton step using the approximate solution u∗i−1 ∈ Hi−1 from the previous grid as initial guess. This leads to the Newton system: Find zi ∈ Hi such that Lu∗i−1 (zi − u∗i−1 , vi ) = −L(u∗i−1 , vi )
for all vi ∈ Vi .
Usually it is rewritten in the form Lu∗i−1 (zi , vi ) = g(u∗i−1 ), vi 0
(12)
for all vi ∈ Vi
with g(u∗i−1 ), vi 0 = −L(u∗i−1 , vi ) + Lu∗i−1 (u∗i−1 , vi ) −f (x, u∗i−1 )vi + ∂2 f (x, u∗i−1 )u∗i−1 vi dx. = Ω
Due to (9) the linear system (12) has a unique solution. It is solved iteratively using u∗i−1 as initial guess. As a result we obtain the approximate solution u∗i . Here we consider the following type of basic iterations Ii,mi for solving a linear system Lr (zi , vi ) = g, vi
for all vi ∈ Hi ,
namely zi − Ii,mi zi0 = Si,mi (zi − zi0 ) where zi0 ∈ Hi is the initial guess, and Si,mi : Hi → Hi is a linear mapping for the error propagation after mi iteration steps. We call an iterative solver a smoother if it satisfies the smoothing properties (13) (14)
c vi L2 (Ω) hi mγi [[Si,mi vi ]]r ≤ [[vi ]]r
[[Si,mi vi ]]r ≤
for all vi ∈ Hi , for all vi ∈ Hi .
Remark 1. As is shown in [Hac85] the symmetric Gauß-Seidel, the SSOR, and the damped Jacobi iteration are smoothers with a parameter γ = 1/2. When using the conjugate gradient method as basic iteration we have to tackle with the problem that its error propagation operator is not linear. However, in [Sha96], [BD96] it is shown that the error propagation operator of the conjugate gradient method is majorized by the linear error propagation operator of a special Jacobi-type iteration which is a smoother in the sense of (13), (14) with γ = 1.
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Now we can formulate the cascadic multigrid algorithm for problem (5). Cascadic multigrid algorithm Initialization:
Determine u∗0 ∈ H0 as an approximate solution of (5) on level i = 0.
Cascadic Loop:
for i = 1, . . . , l do Solve the Newton system (12) on level i by a smoothing iteration Ii,mi u∗i = Ii,mi u∗i−1 .
3. Error analysis In this section we give a detailed error analysis of the cascadic multigrid algorithm. The cascadic multigrid algorithm is defined by the tupel {l, h0 , hl , m1 , . . . , ml } where l, h0 , and hl are related by hl = 2−l h0 . In the sequel we will show that for appropriately chosen setups and sufficiently good start approximations u∗0 it generates an approximate solution u∗l which satisfies the error estimate ul − u∗l H 1 (Ω) ≤ chl uH 2 (Ω) while the number of arithmetic operations is bounded by cNl . The error analysis is done using the energy norm (10). In this section we will use the simplified notation [[v]]i := [[v]]u∗i−1 . Further, we define the quantities ei := [[ui − u∗i ]]i
and di := [[u∗i − u∗i−1 ]]i
for i = 0, . . . , l setting u∗−1 := u∗0 . In this section we will distinguish by enumeration those constants which are referred to in the remainder. Furthermore, we explicitly specify how the constants depend on the nonlinearity of the problem which is characterized by κ. Then all estimates can be considered with respect to the limit κ → 0. It will turn out that in the limit κ = 0 the results are in accordance with the known estimates for the linear case, see [Sha94a], [BD96]. The outline of the analysis is as follows: In Lemma 1 we provide a relation between the energy norms [[.]]i+1 and [[.]]i with respect to two successive
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approximate solutions u∗i and u∗i−1 . Afterwards we prove recursive estimates for ei and di , see Lemma 2 and Lemma 3. With Lemma 4 we provide a condition for bounding the iteration error ei . Theorem 1 establishes the main result for the cascadic algorithm. Lemma 1 The norms [[.]]i+1 and [[.]]i are related by (15)
[[v]]i+1 ≤ (1 + c1 κdi )[[v]]i
for all v ∈ H01 (Ω)
with some constant c1 > 0 which is independent of i. Proof. The proof is straightforward. Using the Taylor expansion ∂2 f (x, u∗i ) = ∂2 f (x, u∗i−1 ) + ∂22 f (x, ξ)(u∗i − u∗i−1 ) with ξ = ξ(x) lying between u∗i−1 = u∗i−1 (x) and u∗i = u∗i (x) and applying the assumptions (2), (3) we obtain [[v]]2i+1
= =
2 Ω j,k=1
2
Ω j,k=1
≤ [[v]]2i + κ ≤
[[v]]2i
ajk ∂k v∂j v + ∂f2 (x, u∗i )v 2 dx ajk ∂k v∂j v + ∂f2 (x, u∗i−1 )v 2 dx
+
Ω
∂22 f (x, ξ)(u∗i − u∗i−1 )v 2 dx
|u∗i − u∗i−1 |v 2 dx
Ω
+ κu∗i − u∗i−1 L2 (Ω) v2L4 (Ω) .
With the embeddings H01 (Ω) '→ L2 (Ω), H01 (Ω) '→ L4 (Ω) and the norm equivalence (11) we get [[v]]2i+1 ≤ [[v]]2i + cκu∗i − u∗i−1 H 1 (Ω) v2H 1 (Ω) ≤ [[v]]2i + c1 κ[[u∗i − u∗i−1 ]]i [[v]]2i
≤ (1 + c1 κdi )[[v]]2i ≤ (1 + c1 κdi )2 [[v]]2i . Taking the square root we end up with (15). The above result is needed whenever terms from a given level have to be expressed in terms of the previous level. In particular, it is used for the recursive estimates of ei and di . In the next lemma we derive such an estimate for the iteration error ei .
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Lemma 2 Consider the cascadic multigrid algorithm with a smoother in the sense of (13), (14) as basic iteration. Then the error ei can be estimated recursively by ei ≤ (1 + c1 κdi−1 )ei−1 + c2 κh2i u2H 2 (Ω) (16)
+ c3 κhi uH 2 (Ω) ei−1 + c4 κe2i−1 + c5
hi uH 2 (Ω) mγi
with appropriate constants ck > 0 which are independent of i. Proof. The error ei consists of two parts, namely one arising due to Newton’s method and another caused by the iterative solution of the linear system (12) ei = [[ui − u∗i ]]i ≤ [[ui − zi ]]i + [[zi − u∗i ]]i . The Newton error can be estimated by ui − zi H 1 (Ω) ≤ cκui − u∗i−1 2H 1 (Ω) , see [Sha95a]. Due to the norm equivalence (11) which is uniform with respect to r we can substitute the H 1 -norm on the left-hand side by [[.]]i and on the right-hand side by [[.]]i−1 which leads to [[ui − zi ]]i ≤ cκ[[ui − u∗i−1 ]]2i−1 . Using the norm equivalence (11), the discretization error estimate (6), and the relation hi−1 = 2hi the right-hand side satisfies [[ui − u∗i−1 ]]i−1 ≤ [[ui − ui−1 ]]i−1 + [[ui−1 − u∗i−1 ]]i−1 ≤ cui − ui−1 H 1 (Ω) + ei−1 ≤ chi uH 2 (Ω) + ei−1 . Hence, the Newton error can be estimated by (17)
[[ui − zi ]]i ≤ cκ(hi uH 2 (Ω) + ei−1 )2 .
The error coming from the iterative solution of the linear system is estimated using the smoothing properties (13), (14) [[zi − u∗i ]]i = [[Si,mi (zi − u∗i−1 )]]i ≤ [[Si,mi (zi − ui )]]i + [[Si,mi (ui − ui−1 )]]i + [[Si,mi (ui−1 − u∗i−1 )]]i c ≤ [[zi − ui ]]i + ui − ui−1 L2 (Ω) + [[ui−1 − u∗i−1 ]]i . hi mγi The first term is again the Newton error which can be estimated as above. For the second term the discretization error estimate (7) and hi−1 = 2hi imply ui − ui−1 L2 (Ω) ≤ ch2i uH 2 (Ω) .
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Applying Lemma 1 to the last term we end up with ei ≤ 2cκ(hi uH 2 (Ω) + ei−1 )2 + c5
hi uH 2 (Ω) + (1 + c1 κdi−1 )ei−1 mγi
which leads to (16) with appropriate constants ck > 0. Next we need an estimate for di−1 . Lemma 3 The quantity di can be estimated by (18)
di ≤ ei + c6 hi uH 2 (Ω) + (1 + c1 κdi−1 )ei−1 .
Proof. We have di = [[u∗i − u∗i−1 ]]i ≤ [[u∗i − ui ]]i + [[ui − ui−1 ]]i + [[ui−1 − u∗i−1 ]]i . Then (18) follows with the estimate (19)
[[ui − ui−1 ]]i ≤ cui − ui−1 H 1 (Ω) ≤ c6 hi uH 2 (Ω)
for the second term and Lemma 1 applied to the last term. Note that the estimates for ei and di are coupled. However, since d0 = 0 we can estimate e1 immediately provided that e0 is given. Then, from e1 we obtain an estimate for d1 which in turn allows to bound e2 , and so on. In other words, starting with e0 and d0 = 0 we can alternately estimate ei and di . Let us rewrite (16) in the following way ei ≤ 1 + κ(c1 di−1 + c3 hi uH 2 (Ω) + c4 ei−1 ) ei−1 (20) hi + c2 κh2i u2H 2 (Ω) + c5 γ uH 2 (Ω) . mi Our intension is to bound ei by a multiple of hl for all i. In particular, the term with h2i has to be bounded by hl for all i. This leads to a condition for h0 . Moreover, we need a uniform bound for an expression of the type (1 + cκh0 )i for i = 1, . . . , l. The next lemma provides such a result. Lemma 4 Let h0 and hl be related by (21)
h20 ≤
c7 hl κ
with some fixed constant c7 ≥ 1. Then the following assertion holds: For any constant c > 0 we can choose h0 and hl in accordance with (21) sufficiently small such that (22)
(1 + cκh0 )l ≤ 2c7 .
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Proof. For any c > 0 the function g(x) = (1 + cx)− log2 x is continuous on (0, ∞) and has the limit limx→+0 g(x) = 1. Hence, there exists a constant b = b(c) > 0 such that (23)
g(x) ≤ 2 for all x ∈ (0, b].
Using hl = 2−l h0 we can rewrite (21) as h0 ≤
c7 2−l κ
which leads to l ≤ log2 c7 − log2 κh0 . Let 1 −l 1 h0 ≤ min c7 2 , , b(c) . κ 2 With (23) and κh0 ≤ b(c) we get (1 + cκh0 )− log2 κh0 ≤ 2 which implies 1 + cκh0 ≤ 2 since − log2 κh0 ≥ 1 for h0 ≤ 1/2κ. Furthermore, we have (1 + cκh0 )log2 c7 ≤ c7 taking into account that c7 ≥ 1. Summarizing these estimates yields (1 + cκh0 )l ≤ (1 + cκh0 )log2 c7 −log2 κh0 ≤ 2c7 . Remark 2. The crucial point is that the term (1 + cκh0 )l can be bounded by a constant which is independent of l, h0 , hl , and c provided that h0 and hl = 2−l h0 are sufficiently small. Now we can formulate the main result for the cascadic multigrid algorithm. We recall that x denotes the smallest integer ≥ x. Theorem 1 Let the basic iteration of the cascadic multigrid algorithm be a smoother in the sense of (13), (14). Choose the number mi of iteration steps at level i according to (24)
mi = ml β l−i
for i = 1, . . . , l
where β > 21/γ is a fixed constant. Suppose that h0 and hl satisfy the relation c7 h20 ≤ hl . κ Let u∗0 be a sufficiently good approximate solution of (5) on level i = 0 such that (25)
e0 ≤ εhl uH 2 (Ω)
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with some constant ε > 0. Then the following assertion holds: For sufficiently small h0 the cascadic multigrid algorithm generates an approximate solution u∗l such that el ≤ c8 hl uH 2 (Ω)
(26) where the constant (27)
c8 := 2c7
1 c5 ε + c2 c7 uH 2 (Ω) + γ 3 ml (1 − 2/β γ )
is independent of l, h0 , and hl . Proof. We define the constants c9 := 3c8 + c6 , c10 := (c1 c9 + c3 + c4 c8 )uH 2 (Ω) and choose (28)
1 1 1 h0 ≤ min c7 2−l , , b(c10 ), . κ 2 c1 c9 uH 2 (Ω)
Observe that with this choice of h0 the assumptions of Lemma 4 are satisfied for c = c10 . Hence, we obtain (29)
(1 + c10 κh0 )i ≤ 2c7
for all i = 1, . . . , l.
We will show that for i = 0, . . . , l i
ei ≤ (1 + c10 κh0 ) e0 +
c2 κu2H 2 (Ω)
i
h2j (1 + c10 κh0 )i−j
j=1
i hj + c5 uH 2 (Ω) (1 + c10 κh0 )i−j , mγj
(30)
j=1
(31)
ei ≤ c8 hl uH 2 (Ω) ,
(32)
di ≤ c9 hi uH 2 (Ω) .
For the proof we use induction on i. Obviously, these estimates are satisfied for i = 0 due to (25) and d0 = 0. Assume that (30)–(32) hold for a
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fixed level i > 0. Then from (20) we obtain ei+1 ≤ 1 + κ(c1 di + c3 hi+1 uH 2 (Ω) + c4 ei ) ei hi+1 + c2 κh2i+1 u2H 2 (Ω) + c5 γ uH 2 (Ω) mi+1 ≤ 1 + κ(c1 c9 + c3 + c4 c8 )uH 2 (Ω) h0 ei hi+1 + c2 κh2i+1 u2H 2 (Ω) + c5 u2 γ mi+1 ≤ (1 + c10 κh0 )i+1 e0 + c2 κu2H 2 (Ω) + c5 uH 2 (Ω)
i+1
h2j (1 + c10 κh0 )i+1−j
j=1 i+1 hj (1 + c10 κh0 )i+1−j mγj j=1
which is (30) for i + 1. Here we used hi−1 = 2hi , (24), and the estimates (30)–(32) on level i. Applying (29) we can continue with ei+1 ≤ 2c7 e0 + 2κc2 c7 u2H 2 (Ω)
i+1
h2j + 2c5 c7 uH 2 (Ω)
j=1
i+1 hj . mγj j=1
Moreover, hi−1 = 2hi , κh20 ≤ c7 hl , and (24) lead to ei+1 ≤ 2c7 e0 +
2c2 c7 κu2H 2 (Ω) h20
i+1
4−j
j=1
i+1 hl 2 l−j + 2c5 c7 uH 2 (Ω) γ βγ ml j=1
1 c5 hl uH 2 (Ω) ≤ 2c7 ε + c2 c7 uH 2 (Ω) + γ 3 ml (1 − 2/β γ ) = c8 hl uH 2 (Ω) .
Finally, Lemma 3, hi ≤ h0 ≤ (c1 c9 κuH 2 (Ω) )−1 , and the estimates (31), (32) on level i imply di+1 ≤ ei+1 + c6 hi+1 uH 2 (Ω) + (1 + c1 κdi )ei ≤ c8 hl uH 2 (Ω) + c6 hi+1 uH 2 (Ω) + 1 + c1 c9 κuH 2 (Ω) hi c8 hl uH 2 (Ω) ≤ 3c8 hl uH 2 (Ω) + c6 hi+1 uH 2 (Ω) ≤ c9 hi+1 uH 2 (Ω) .
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Let us discuss the restrictions on h0 and hl which are caused by the nonlinearity of the problem. The crucial condition is (21) which is necessary to ensure convergence of the cascadic Newton method, in general. It allows to control the right-hand side of the Newton error estimate (17). Moreover, this condition describes how h0 and hl have to be related in dependence on the nonlinearity of the problem: The larger κ, the smaller h0 has to be chosen and accordingly smaller is hl . To illustrate this we consider the following situation. If we prescribe an arbitrary final grid size hl by some a priori considerations concerning the desired error tolerance for el then it is possible that no h0 exists which satisfies (21). For example, let c7 = 1, κ = 1000, and hl = 0.001. Then we have the condition h0 ≥ 2hl = 0.002 since a multigrid method uses at least two levels. On the other hand, (21) requires
hl = 0.001. h0 ≤ κ Consequently, for a problem with such large κ a final grid size hl = 0.001 is not small enough to ensure convergence of Newton’s method. The key question is: For given κ how l, h0 , and hl have to be chosen such that all conditions are satisfied? For answering this question we rewrite (28) as 1 min{c7 2−l , c11 } κ
h0 ≤ with
c11 := min
1 1 , b(c10 ), 2 c1 c9 uH 2 (Ω)
.
To ensure hl < h0 we choose l ≥ 1 such that c7 2−l ≤ c11
or
l ≥ log2
c7 . c11
This l determines h0 ≤ c7 2−l /κ and hl = 2−l h0 ≤ c7 4−l /κ. Summarizing these considerations in the nonlinear case h0 and hl cannot be chosen independently of each other. In particular, this means that h0 and hl have to be chosen the smaller the larger the nonlinearity κ is. Moreover, increasing the number of levels l or reducing hl results in a reduction of h0 , too. However, hl tends to zero faster than h0 due to (21). In the linear case κ = 0 the ratio 1/κ in (28) tends to infinity, i.e., there is no restriction on the initial grid size h0 . This is in accordance with the results for cascadic multigrid algorithms for linear problems, see [Sha94a], [BD96]. Now we want to estimate the arithmetic complexity of the cascadic multigrid algorithm. The amount of work on the levels 1 to l can be estimated as
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in [BD96] since on each level exactly one linear system has to be solved. With Nl ≥ 4l−i Ni , Ni = dim Hi this gives comp(algo) = c12
l i=1
mi N i ≤
c12 ml Nl 1 − β/4
for β < 4. Additionally, we have to take into account the computational 1/2 costs at level 0 since h0 = O(hl ) tends to zero. Therefore, we assume that on the coarsest grid a Newton iteration with conjugate gradient method for the solution of the linearized systems is implemented. The mesh independence principle implies that there will be only a fixed finite number of Newton iterations on a sufficiently small coarse grid to achieve a given tolerance, see [ABPR86]. Taking into account the restriction (25) and the quadratic rate of convergence of Newton’s method the number of Newton steps is O(ln l). In 2D the solution of one Newton system using the conju3/2 gate gradient method without preconditioning costs about O(N0 ) so that 3/2 1/2 the total costs at level 0 can be estimated by O(N0 ln l). With N0 ∼ Nl 3/4 this gives O(Nl ln l) which is less than O(Nl ). Thus, the computational complexity of the cascadic multigrid algorithm is O(Nl ) and we get both accuracy and multigrid complexity as long as 21/γ < β < 4 that is for γ > 1/2. Finally, we consider the constant c8 defined in (26) in more detail. In the linear case the error el can be estimated by el ≤
c hl uH 2 (Ω) mγl (1 − 2/β γ )
if β > 21/γ , see [BD96]. This implies that the final error decreases proportionally to mγl . In the nonlinear case there is an additional influence of the Newton error and the initial error. These errors are controlled by the conditions on h0 and u∗0 . Their influence decreases if κ → 0, and in the limit κ = 0 we obtain the same result as in [BD96]. Acknowledgements. The author would like to thank Prof. V. V. Shaidurov for fruitful discussions, in particular during his stay at the Technical University of Dresden in October 1997. This co-operation was supported by the Deutsche Forschungsgemeinschaft under grant GR 705/4-2. Moreover, the author is grateful to Prof. H. Schwetlick for support and supervision. Finally, the author acknowledges the improvements made possible by the comments of Prof. F. A. Bornemann.
A cascadic multigrid algorithm for semilinear elliptic problems
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