MULTIGRID PRECONDITIONING FOR THE BIHARMONIC DIRICHLET PROBLEM M. R. HANISCH

MULTIGRID PRECONDITIONING FOR THE BIHARMONIC DIRICHLET PROBLEM M. R. HANISCH Abstract. A multigrid preconditioning scheme for solving the Ciarlet-Raviart mixed method equations for the biharmonic Dirichlet problem is presented. In particular, a Schur complement formulation for these equations which yields non-inherited quadratic forms is considered. The preconditioning scheme is compared with a standard W-cycle multigrid iteration. It is proved that a Variable V-cycle preconditioner leads to problems with uniformly bounded condition numbers. However, W-cycle convergence is proved only if the number of smoothings \m is suciently large". An example is given in which the W-cycle diverges unless m 8. Divergent W-cycles are also encountered when solving the Morley equations for the biharmonic Dirichlet problem; although, Brenner has proved W-cycle convergence for suciently large m [9]. This is illustrated with additional computations, while Variable V-cycles continue to produce excellent preconditioners in this setting. Certain approximate L 2-inner products are described and a modication to the Ciarlet-Raviart method is proposed which reduces the wor of the multilevel schemes. Optimal order error estimates are proved for the modied method. Consideration is restricted to Ciarlet-Raviart methods of quadratic and higher degree throughout the paper. 1. Introduction. In this paper we introduce and analyze the behavior of multigrid iterative schemes for solving the Ciarlet-Raviart mixed method equations (2.7) for the biharmonic Dirichlet problem in the plane. Other mixed methods for the same problem, the methods of Herrmann and Miyoshi [20, 21, 27], and Herrmann and Johnson [20, 21, 22], and the method of Raviart and Thomas [31] for second-order problems, may be analyzed using the same techniques. For each of these methods, the approximate solution satises an ill-conditioned and indenite bloc matrix equation of the form (1.1) N h uh! = A Bt B?C! h uh! =?d?f For the Ciarlet-Raviart method uh and h are vectors of approximate nodal values for the solution and its negative Laplacian. For simplicity we shall tae d = 0. In this case (1.1) corresponds to the homogeneous Dirichlet problem. The matrix N doesn't change with the boundary data and the results of this paper will extend immediately to non-homogeneous Dirichlet problems. Eliminating h from (1.1), with d = 0 one obtains an equation involving the illconditioned but typically symmetric positive denite Schur complement! (1.2) S uh = [BA?1 B t + C] uh = f It will be proved that a multigrid W-cycle iteration which uses a suciently large number of smoothing steps, \m is suciently large" in the language of Theorem 6, can be used to solve (1.2). The hypothesis that \m is suciently large" is common to many multigrid analyses. In practical computations, \m = 1" is suciently large for many problems. However, this will not be true for (1.2). (Nor will this necessarily be true for the W-cycle THIS WORK WAS PARTLY SUPPORTED BY THE U.S. ARMY RESEARCH OFFICE THROUGH THE MATHEMATICAL SCIENCES INSTITUTE OF CORNELL UNIVERSITY. 1

of Brenner [9] for the Morley discretization [28] of the biharmonic Dirichlet problem, see Section 6.) Alternatively, it will be possible to construct a multigrid preconditioner B for the Schur complement S. (A multigrid preconditioner may similarly be constructed for the Morley equations.) Iterative schemes for the preconditioned problem (1.3) BSuh = Bf (e.g. conjugate gradients) are nown to converge with a rate which is bounded by a function of the condition number of BS, 2 (BS). The smaller the condition number, the better the bound, as we shall see in (3.15) of Section 3.2. It will be shown that a \Variable V-cycle" multigrid preconditioner yields problems (1.3) with small condition number, bounded independently of the mesh diameter for a scale of underlying nite element meshes. We shall then say that B and S?1 are spectrally equivalent, or that BS is spectrally equivalent to the identity. One can construct a variety of wea formulations for the biharmonic Dirichlet problem which are of the abstract form (1.4) 8 >< > Given real Banach spaces V and W, and a (; ), b (; ), and c (; ), bilinear forms on V V, V W, and W W respectively, Find f; ug 2 V W such that for f 2 W 0 ; a (; v) + b (v; u) = 0; 8 v 2 V ; b (; w)? c (w; u) =?(f; w); 8 w 2 W The mixed method of Ciarlet and Raviart [14] is a nite element method for one such formulation. Denote the nite element approximate solution pair by f h ; u h g 2 V h W h, where V h and W h are (properly chosen) nite dimensional subspaces of V and W constructed with respect to a domain triangulation with mesh diameter h. Given bases f' i g and f i g for V h and W h, the mixed method equations may be written in matrix form (1.1) with [A] ij = a (' i ; ' j ) ; [B] ij = b (' j ; i ) ; and [f] j = (f; j ) ; where h and uh are the coecient vectors of h and u h. The resulting matrix N is indenite and extremely ill-conditioned for ne triangulations; in fact, 2 (N) = O(h?4 ). The Schur complement S is similarly ill-conditioned, but is symmetric positive denite. It has been shown by Peiser in [29] that a multigrid W-cycle iteration for N 2 = N t N, the normal equations, converges if \m is suciently large". We shall tae a dierent approach, applying multigrid iterative schemes to the equation (1.2). It is important therefore to observe that for the Ciarlet-Raviart method, A is an L 2 -Gramm matrix. In principle then, the action of A?1 may be computed with a rapidly converging iteration; although, a method will be described for avoiding this iteration. With uh obtained from (1.2), one may then compute h =?A?1 B t uh. A multigrid analysis based on the so-called \non-inherited form" theory of Bramble, Pascia, and Xu [8] is provided for (1.2). Accordingly, an \Approximation and Regularity" property will be proved, see Section 4.2. For this analysis, H 3 -regularity will be assumed for the biharmonic Dirichlet problem; i.e. that the problem is posed on a convex polygonal domain. The multigrid analysis is extended to nonconvex polygonal domains in [19]. 2

Once the Approximation and Regularity property is veried, it is possible to prove that a multigrid W-cycle iteration for (1.2) converges if \m is suciently large". In practice, W-cycle iterations for (1.2) typically diverge if m is small. A numerical example is given in Section 6 for which W-cycle convergence is not guaranteed unless m 8. (This cannot happen in the \inherited form, nested space" setting [2, 3, 6, 24, 25, 26].) It will be preferable then to view the eect of a single multigrid iteration as a preconditioner for the Schur complement S. Constructing a preconditioner from a Variable V-cycle multigrid iteration, it will be seen in Section 3.2 that preconditioned problems with small and uniformly bounded condition numbers can be obtained. This is an improvement over a result of Braess and Peiser [4]. Furthermore, the cost of applying the preconditioner is comparable to that of a single W-cycle iteration with m = 1. The problem (1.2) may then be rapidly solved with a preconditioned conjugate gradient iteration. Consider now the problem of computing the action of BA?1 B t. For the Raviart- Thomas mixed method, in the special case of a rectangular \triangulation" with tensor product elements, A can be cheaply inverted. However, the cheapest procedure for computing the action of BA?1 B t when A is well-conditioned generally involves an inconvenient second iteration. We will show how one can avoid this \inner iteration" for the Ciarlet- Raviart method when quadratic or cubic elements are used. It is possible to approximate a (; ) with a new bilinear form a h (; ), in a sense to be made more precise in the sequel, so that the solution f d;h ; u d;h g 2 V h W h of the system (1.5) a h ( d;h ; v) + b (v; u d;h ) = 0; 8 v 2 V h ; b ( d;h ; w)? c (w; u d;h ) =?(f; w); 8 w 2 W h ; approximates f; ug. Associated with a h (; ) will be a new basis f ~' i g for V h, which is also local, and a Gramm matrix [A d ] ij = a h ( ~' i ; ~' j ) which is diagonal. For the examples, this amounts to approximating the L 2 inner product with certain discrete inner products. With respect to the basis for W h and the new basis for V h, (1.6) S d ud;h = [B d A?1 d Bt d + C] ud;h = f One may then iteratively solve for ud;h. The multigrid analysis will apply equally well to (1.6). Alternatively, one may mae use of the fact that a preconditioner B d developed for S d also preconditions S. This will be a consequence of the spectral equivalence of A d and A. Consider the following technique of Bramble and Pascia [7]. Let A 0 be a preconditioner for A which is also \smaller", A 0 A, or ha 0 v; vi hav; vi ; for all vectors v, where h; i denotes the Euclidean inner product. For example, tae A 0 to be the properly scaled diagonal of A. The equation (1.1) may be rewritten (1.7) M h uh! = A 0?1 A A?1 0 Bt BA?1 0 (A? A 0) BA?1 0 Bt + C! h uh! ) = 0 f The matrix M is then symmetric and positive denite with respect to the inner product [[; ]] dened by ""!!## ' ; ha'; i? ha u v 0 '; i + hu; vi 3!

Consider then the matrices ~M I 0 0 BA?1 B t + C! and ~ B I 0 0 B d! Bramble and Pascia prove that ~ M is spectrally equivalent to M. Consequently, ~ B may be used to precondition ~ M and therefore M. One may also use the Variable V-cycle preconditioner constructed for S d to improve the method of Peiser and Braess [30] for solving the Morley nonconforming nite element equations for the biharmonic equation. We mention again the analysis provided by Brenner in [9] for multigrid applied directly to the Morley equations on convex polygons. This paper is arranged in the following manner. In Section 2, we provide an outline of the Ciarlet-Raviart method for the biharmonic Dirichlet problem. We prove error estimates for a perturbed version of this method which will be used in the multigrid analysis. The multigrid preconditioner is considered in Section 3, and in Section 4 we prove the Approximation and Regularity condition. In Section 5 we describe several forms a h (; ) which approximate a (; ) and which yield operators A d which are diagonal in certain local bases for V h. In the nal section, we present the results of several computations. Throughout this paper we use C to denote a generic positive constant which is independent of the mesh parameter h. 2. A Model Problem. In this section we provide a description of the Ciarlet-Raviart method for approximately solving the biharmonic Dirichlet problem (2.1) on convex polygonal planar domains. We also describe certain perturbations by quadrature of this standard Ciarlet-Raviart discretization. These perturbed methods will retain the optimal accuracy of Ciarlet-Raviart solutions, while the quadratures reduce the expense of the multigridpreconditioned iterative methods considered in this wor. In the course of this section we highlight those features of the discretizations which lead to optimal error estimates and which provide a basis for the multigrid analysis of Section 3. Similar features are exhibited by the Herrmann-Miyoshi and Herrmann-Johnson methods for the biharmonic Dirichlet problem and by the Raviart-Thomas method for the second-order Dirichlet problem. A multigrid analysis for these methods is given in [19]. Given a convex polygonal domain in R 2 with boundary @, consider the model problem (2.1) 4 2 u = f in ; u = @u @ = 0 on @; where 4 denotes the Laplacian operator and @ @ is the normal derivative at the boundary of. This problem provides a simple model for the displacement of a clamped elastic plate, or for the stream function of a steady-state planar Stoes ow. Recall the denition, for non-negative integer s, (2.2) H s () def = fv 2 L 2 () D v 2 L 2 (); for jj sg ; where D denotes the distributional derivative, and is a multi-index. The Sobolev space 4

H s () is a Hilbert space with norm, v s; = 0 @ X jjs D v 2 L 2 () Additional spaces H0 s() may be dened as the completions of C1 0 () with respect to the norms s;. (Denote by C 1 0 () the space of innitely dierentiable functions with compact support contained in.) We shall also mae reference to the negative norm (dual) spaces H?s () def = [H0 s()]0. With f 2 H?1 (), the unique existence of a solution u 2 H 3 () \ H0() 2 to (2.1) is nown [18]. We also have the regularity result for convex polygonal domains, (2.3) u 3 C f?1 ; for a constant C which is independent of f, see [18]. In the remainder of this paper we shall assume that is a convex polygon. The case of nonconvex polygonal domains, for which (2.3) does not in general hold, is considered in [19]. (2.4) 2.1. The Ciarlet-Raviart method. Introducing an auxiliary variable such that 1 A 1=2 =?4u and? 4 = f ; and using a standard Green's formula, one may obtain a wea formulation for (2.1) of the type (1.4) with c (; ) 0 (2.5) 8 >< > Find f; ug 2 H 1 () H 1 0() such that for f 2 H?1 (); (; v) L2? D (v; u) = 0; 8 v 2 H 1 ();? D (; w) =?(f; w); 8 w 2 H0 1 () ; where the \Dirichlet form", D (; ), and the L 2 -inner product are given by D ('; v) def = Z r' rv dx ' 1 v 1 ; ( ; ') L 2 def = Z ' dx This formulation was studied by Ciarlet and Raviart in [14]. It is not dicult to show that H.1 The problem (2.5) has a unique solution f; ug 2 H 1 () H 1 0 () for all f 2 H?1 (). Furthermore, =?4u, and this same u solves (2.1) in an appropriate sense. Given a regular and quasi-uniform triangulation (in the sense of Ciarlet [13]) h of, with mesh diameter h, dene nite dimensional subspaces V h H 1 () and W h H 1 0() from the space (2.6) S m h def = fv 2 C 0 () vj T 2 P (m) (T ); 8 T 2 h g ; where P (m) (T ) is the space of polynomials of degree m or less over triangle T. Set V h = Sh m and W h = Sh m \ H1 0 (). The Ciarlet-Raviart mixed method approximates the solution f; ug of (2.5) with the solution to the following problem (2.7) 8 >< > Find f h ; u h g 2 V h W h such that for f 2 H?1 (); ( h ; v) L2? D (v; u h ) = 0; 8 v 2 V h ;? D ( h ; w) =?(f; w); 8 w 2 W h 5

Choosing bases f' i g for V h and f j g for W h, consider the linear system (2.7) in bloc matrix form with the notations [B h ] ij =?D (' j ; i ), [A h ] ij = (' i ; ' j ) L2, [f] j = (f; j ), and denote the transpose of B h by B t h. Applying bloc Gaussian elimination as discussed in the Introduction, one obtains the reduced system for (the coecients of) u h (2.8) B h A?1 h Bt h u h = f In order to avoid computing the action of the Gramm matrix A?1 h, we shall introduce an additional approximation. In particular, we hypothesize the existence of, and in Section 5 construct, bilinear forms (; ) h satisfying H.2 The bilinear form (; ) h is symmetric, and there is a constant > 0 (independent of the mesh parameter h) such that (v; v) h v 0 2 ; 8 v 2 Vh As the form (; ) h is destined to replace the L 2 inner product in (2.7), an impending error analysis will require certain bounds for the quadrature error (2.9) E h (u; v) def = (u; v) h? (u; v) L2 ; for u; v 2 V h We postpone a precise description of the required bounds for E h (; ); however, see (2.18) and (2.19). Using the approximate form leads to a new problem (2.10) 8 >< > Find f d;h ; u d;h g 2 V h W h such that for f 2 H?1 (); ( d;h ; v) h? D (v; u d;h ) = 0; 8 v 2 V h ;? D ( d;h ; w) =?(f; w); 8 w 2 W h We then consider the associated reduced system (2.11) B d;h A?1 d;h Bt d;h u d;h = f ; where A d;h is the Gramm matrix associated with (; ) h. In Section 5 we shall construct forms (; ) h satisfying H2 which yield operators A d;h which are diagonal if an appropriate local basis is chosen for V h. Unique existence of solutions for the problem (2.7) was proved in [14]. Alternatively, Fal and Osborn show that unique existence is essentially a consequence of the property H4 described in the next subsection. Using the coercivity from H2, the proof given in [15] is easily modied to yield Theorem 1. If H1, H2, and H4 are valid, then (2.10) has a unique solution. 2.2. Error estimates. Before proving error estimates for the solution f h ; u h g of (2.10) we mae two more observations. First, a well-nown consequence of the quasiuniformity of the triangulations h is H.3 (Inverse Property) There is a constant C which does not depend on h such that A second observation is v 1 C h?1 v 0 ; 8 v 2 V h 6

H.4 There is an operator h H 1 ()! V h such that D (v? h v; w) = 0; 8 w 2 W h In fact, we may tae the operators h to be Neumann projections; that is, for u 2 H 1 (), h u 2 V h ; D ( h u; v) = D (u; v); 8 v 2 V h ; and Z h u dx = Z u dx Since W h V h, it follows immediately that H4 is satised. Furthermore, for v 2 H r?2 () with r 3, the following estimates are nown, (2.12) v? h v j Ch l?j v l ; j = 0; 1; 1 l min(m + 1; r? 2) Remar 2. Taen together, the observations H2 and H4 are closely related to the so-called \inf-sup conditions" which were shown by Babusa [1] and Brezzi [10] to lead to unique existence, and error estimates, for (2.5). For a discussion of this relationship, we refer the reader to the papers of Fal and Osborn [15] and Fortin [16]. 2 In [15] it is shown that, by manipulating the equations (2.5) and (2.7), estimates for the errors? h 0 and u? u h 1 may be obtained from approximation properties of the operators h and the spaces W h. Similarly, we may obtain the following intermediate error estimates for the problem (2.10) Theorem 3. If H1, H2 and H4 are valid, and if f; ug and f d;h ; u d;h g denote the solutions of (2.5) and (2.10) respectively, then (2.13) u? u d;h 1 = sup d2h?1 ()nf0g? D (y d? h y d ; u? w)? D (? d;h ; d? ) + (? d;h ; y d? h y d ) L2 + E h ( d;h ; h y d ) / d?1 ; 8 w; 2 W h where fy d ; d g 2 H 1 () H0 1() is the solution of (2.5) with?f replaced by d 2 H?1 (), and E h (; ) is dened by (2.9). If additionally H3 is valid, then? d;h 0 1 n (2.14) C h?1 u? w 1 where E h () is dened by + (1 + )? h 0 + E h () o ; 8 w 2 W h ; (2.15) E h () def je h ( h ; v)j = sup v2v h nf0g v 0 7

We postpone the proof of this theorem until Section 2.3. Recalling the regularity (2.3), assume that the solution u of (2.5) is H s ()-regular for some s 3. If the spaces W h are obtained from Sh m with m s? 1, the following approximation property is obtained, inf w2w h u? w 1 Ch s?1 u s for all u 2 H 1 0() \ H s (). Combining this result with (2.12), (2.14) becomes (2.16)? d;h 0 Ch s?2 u s + E h () since s?2 = 4u s?2 u s. Estimating the terms in (2.13), inf w2w h D (y d? h y d ; u? w) C y d? h y d 1 inf w2w h u? w 1 C y d 1 Ch s?1 u s ; (? d;h ; y d? h y d ) L2? d;h 0 y d? h y d 0 [ C h s?2 u s + E h () ] C h y d 1 If one further assumes that E h () Ch s?2 u s, compare (2.18), then inf D (? d;h ; d? ) [? h 2W 1 + C h?1 h? d;h 0 ] inf d? h 2W 1 h C h s?3 u s Ch 2 d 3 ; using H3, for m 2. Recalling the a priori inequality y d 1 d 3 C d?1, one consequently obtains the following estimate from (2.13), (2.17) u? u d;h 1 Ch s?1 u s + sup d2h?1 ()nf0g E h ( d;h ; h y d ) ; if E h () Ch s?2 u d s?1 If we tae (u; v) h (u; v) L2, then the last bounding terms in (2.16) and (2.17) vanish, giving the estimates which were obtained by Fal and Osborn for the Ciarlet-Raviart discretization (2.7). In order to match with f d;h ; u d;h g the accuracy of the solution f h ; u h g to (2.7) we shall require that E h (; ) satisfy (2.18) and (2.19) E h () = je h ( h ; v)j sup Ch s?2 u v2v h nf0g v s ; 0 E h ( d;h ; h y d ) sup Ch s?1 u d2h?1 ()nf0g d s?1 Forms (; ) h for which these conditions are satised are given in Section 5. Remar 4. The estimates (2.16) and (2.17) are not proved for s < 3. Indeed, these estimates rely on (2.12) with l = s? 2, and therefore l 1. Since we have taen m s? 1, the estimates (2.16) and (2.17) are valid only for piecewise quadratic or higher order elements. 2 8

2.3. Proof of Theorem 3. We rst prove (2.14). Subtracting the equations (2.10) from the corresponding equations in (2.5) we obtain (2.20) (2.21) (; v) L2? ( d;h ; v) h? D (v; u? u d;h ) = 0; 8 v 2 V h ; D (? d;h ; w) = 0; 8 w 2 W h Property H4 and (2.21) yield (2.22) Adding ( h ; v) L2 D ( h? d;h ; w) = 0; 8 w 2 W h to both sides of (2.20) and rearranging gives (2.23) ( h ; v) L2? ( d;h ; v) h = D (v; u? u d;h ) + ( h? ; v) L2 Setting v = h? d;h in (2.23) and applying (2.22), ( h ; h? d;h ) L2? ( d;h ; h? d;h ) h = D ( h? d;h ; u? w) + ( h? ; h? d;h ) L2 ; 8 w 2 W h ; hence, (2.24) ( h? d;h ; h? d;h ) h = ( h ; h? d;h ) h? ( h ; h? d;h ) L2 + D ( h? d;h ; u? w) + ( h? ; h? d;h ) L2 ; 8 w 2 W h From H2, (2.24), and H3, and recalling the denition (2.9) of E h (; ), h? d;h 0 2 ( h? d;h ; h? d;h ) h E h ( h ; h? d;h ) + D ( h? d;h ; u? w) + ( h? ; h? d;h ) L2 E h ( h ; h? d;h ) + d;h? h 1 u? w 1 + h? 0 h? d;h 0 je h ( h ; h? d;h )j + Ch?1 u? w h? d;h 1 0 +? h 0 h? d;h 0 Hence, (2.25) h? d;h 0 1 E h () + Ch?1 u? w 1 +? h 0 ; 8 w 2 W h ; and (2.14) follows from the triangle inequality. To prove (2.13) we apply H1 as in [15] to obtain (2.26) u? u d;h 1 = (d; u? u d;h ) sup = sup d2h?1 ()nf0g d?1 d2h?1 ()nf0g 9? D (y d ; u? u d;h ) d?1

Then, using H4 (2.27) D (y d ; u? u d;h ) = D ( h y d ; u? u d;h ) + D (y d? h y d ; u? u d;h ) = D ( h y d ; u? u d;h ) + D (y d? h y d ; u? w) ; 8 w 2 W h Using (2.20), (2.21) and the denitions of fy d ; d g and h, for all 2 W h (2.28)? D ( h y d ; u? u d;h ) = ( d;h ; h y d ) h? (; h y d ) L2 = ( d;h ; h y d ) h? (; h y d? y d ) L2? (; y d ) L2 + ( d;h ; y d ) L2? ( d;h ; y d ) L2 = ( d;h ; h y d ) h? (; h y d? y d ) L2 Combining (2.26), (2.27) and (2.28) yields (2.13).? D (? d;h ; d )? ( d;h ; y d ) L2 = ( d;h? ; h y d? y d ) L2? ( d;h ; h y d ) L2 + ( d;h ; h y d ) h? D (? d;h ; d? ) = (? d;h ; y d? h y d ) L2 + E h ( d;h ; h y d )? D (? d;h ; d? ) 3. Multigrid Algorithms. In this section we will describe several multigrid approaches for solving the mixed method discretizations presented in the previous section. A standard multigrid W-cycle is considered rst; namely, Algorithm 3.1 in which p is set equal to 2. It will be shown that this multigrid iteration converges provided that a suciently large but a priori unnown number of smoothing iterations are performed for each multigrid iterative step. In the language of Theorem 6, one must have that \m is suciently large". This indeterminacy with respect to the smoothing is common to many multigrid analyses. In practice, taing the minimal number of smoothing steps (m = 1) leads to W-cycle convergence for many problems. However, for the mixed methods considered in this wor, numerical experiments described in Section 6 show that the W-cycle iteration diverges unless many smoothing iterations are performed with respect to Theorem 6, one may be required to tae m 8. Even if a suitable value for m can be determined, the requirement that m be large leads to an inecient W-cycle iteration as we shall see. For the Ciarlet-Raviart mixed method (and for a non-conforming method of Morley), we shall therefore consider a slightly dierent multigrid approach. In Section 3.1 we shall construct a preconditioner from a single multigrid iteration. Actually, the multigrid iteration is itself nothing more than a preconditioned linear iteration, cf. (3.5) { (3.8). It can be shown that for the socalled \Variable V-cycle" multigrid preconditioner, a uniformly bounded conditioning is obtained, and with no parameters to estimate. Compare the result proved in Theorem 7. Furthermore, the cost of applying this preconditioner is comparable to the cost of a single W-cycle iteration with m = 1. One may then use this preconditioner to obtain a rapidly converging conjugate gradient iteration. This will be illustrated in Section 6. 3.1. Overview. In the next paragraphs we shall describe several multigrid algorithms in an abstract setting. The application of these algorithms to the mixed methods of Section 2.1 will be described in Section 3.3. The algorithms are dened with respect to a 10

sequence of Hilbert spaces W, = 1;...; j, with inner products denoted by (; ), (not to be confused with (; ) h of H2). In general, the spaces W need not be nested; although, for the Ciarlet-Raviart problems W?1 W. For each space W, let A W! W denote a symmetric (self-adjoint with respect to the inner product for W ) positive denite operator. The goal of the multigrid algorithms will be to solve a \ne level" problem, (3.1) A j u j = f j ; using the coarse level operators A. To dene a multigrid algorithm, assume further that one has linear mappings, \prolongations", I W?1! W. For example, if the spaces W are nested, I may be taen to be the natural injection operator. (See [9] for a suitable I in the Morley setting.) Dene \restriction" operators P?1 W! W?1 satisfying (3.2) (P?1 u; v)?1 = (u; I v) ; 8 v 2 W?1 The computability of the adjoint operators P?1 will be considered in the next section. Finally, we shall employ linear \smoothing iterations" associated with the problems A u = f, for = 2;...; j. We postpone the discussion of the suitability of a smoothing iteration, but now claim that point, line, or bloc Jacobi or Gauss-Seidel iterations, or the Richardson iteration, may be eectively used. These smoothing iterations can be expressed in terms of operators R W! W in the following way, (3.3) x i = x i?1 + R (f? A x i?1 ) As the operators R need not be self-adjoint, consider also the \symmetrized" smoothing iterations (3.4) x i = x i?1 + R (i) (f? A x i?1 ) ; where ( R (i) R if i is odd, = R t if i is even, and the superscript t denotes the adjoint/transpose with respect to (; ). For the Gauss- Seidel iteration, using R t in place of R corresponds to reversing the ordering of unnowns, or reversing the sweep direction. One may now dene a symmetric multigrid process for iteratively solving (3.1). Given an initial approximation z l?1 to the solution u of the problem A u = f, compute an improved approximation in W, z l = Mg (z l?1 ; f ). The procedure Mg (; ) is dened below by the recursive Algorithm 1. Setting p = 2 in this algorithm and using m() = m smoothings on each level yields a multigrid W-cycle. With p = 1, a V-cycle is obtained. It is possible to increase the number of smoothings m() as decreases without signicantly increasing the cost required to compute Mg (; ). We shall refer to a Variable V-cycle as that scheme obtained from Algorithm 1 with p = 1 and 0 m() m(? 1) 1 m() and 1 < 0 1 11

procedure Mg (z l?1 ; f ) if = 1, solve exactly return Mg 1 (z1 l?1 ; f 1 ) = A?1 1 f 1 else, dene Mg (z l?1 ; f ) in terms of Mg?1 (; ) as follows 0. initialize x 0 = z l?1 and q 0 = 0 2 W?1 1. smooth m() times x i = x i?1 + R (i) (f? A x i?1 ); i = 1;...; m() 2. perform a coarse grid correction where y m() = x m() + I q p q j = Mg?1 (q j?1 ; P?1 (f? A x m() )); j = 1;...; p 3. smooth m() more times y i = y i?1 + R (i) (f? A y i?1 ); i = m() + 1;...; 2m() return Mg (z l?1 ; f ) = y 2m() Algorithm 1. A symmetric multigrid procedure. Observe that the Variable V-cycle with m() = 2 j? and the W-cycle with m = 1 require the same number of smoothing iterations for each level. Typically a multigrid solution of (3.1) generates iterates (3.5) z l = Mg j (z l?1 ; f j ) from an initial guess z 0. Denoting the l th error e l = u j? z l, it can be shown that e l = E j e l?1 holds for a linear error reduction operator E j W j! W j. Dene a new operator B j associated with the multigrid process by (3.6) One may then rewrite (3.5) in the form B j = (I? E j )A?1 j or E j = I? B j A j (3.7) z l = z l?1 + B j (f j? A j z l?1 ) But this is nothing more than a linear iterative scheme for solving the preconditioned system (3.8) B j A j u j = B j f j 12

Since Algorithm 1 employs a symmetrized smoothing iteration, it can be shown that B j is symmetric with respect to (; ) j, see [8]. Consequently, B j A j is symmetric with respect to the inner product (A j ; ) j. The multigrid iteration (3.5) is contracting provided that the eigenvalues of the symmetric operator E j are contained in the interval (?1; 1), or if E j < 1. Equivalently, the linear iterative scheme (3.7) converges provided that the eigenvalues of B j A j are contained in (0; 2). If the maximum eigenvalue of B j A j is larger than 2, then (3.7) generally diverges; although, a dierent iterative scheme for solving (3.8), conjugate gradients for example, may rapidly converge. This is an important observation for the multigrid iteration when it is applied to the mixed methods of Section 2.1. Indeed, the W-cycle with m = 1 diverges for these methods. However, the problem (3.8) obtained for the Variable V-cycle with p = 1 and m() = 2 j? has a small condition number independent of the mesh size h and is rapidly solved by a conjugate gradient iteration. Even when E j < 1 so that the multigrid iteration (3.5) converges, the spectral condition number of B j A j satises 2 (B j A j ) = max(b j A j ) min (B j A j ) 1 + 1? and it is often preferable to solve (3.8) with a conjugate gradient iteration. It remains to show that the operators B j are computable. Comparing (3.5) and (3.7) it is clear that B j g = Mg j (0; g). Alternatively, operators B may be directly dened by Algorithm 2. if = 1, solve exactly dene B 1 = A?1 1 else, dene B g in terms of B?1 as follows 0. initialize x 0 = 0 2 W and q 0 = 0 2 W?1 1. smooth m() times x i = x i?1 + R (i) (g? A xi?1 ); i = 1;...; m() 2. perform a coarse grid correction where y m() = x m() + I q p q j = q j?1 + B?1 hp?1 (g? A xm() )? A?1 q j?1i ; j = 1;...; p 3. smooth m() more times y i = y i?1 + R (i) (g? A yi?1 ); i = m() + 1;...; 2m() dene B g = y 2m() Algorithm 2. A symmetric multigrid preconditioner. 13

3.2. Convergence theory. We now outline the convergence theory for Algorithms 1 and 2 following the approach taen in [8]. The analysis is based upon two conditions. Before introducing the rst condition, which concerns the smoothing iteration, it is convenient to dene an error operator K = I? R A and its adjoint with respect to the inner product (A ; ), K = I? R t A. C.1) There is a constant C R independent of such that the smoothing procedure satises (3.9) jjjujjj 2 0; C R (R u; u) ; 8 u 2 W ; for both R = (I? K K )A?1 and R = (I? K K )A?1, where is the largest eigenvalue of A, and jjjujjj 0; denotes the norm induced by the inner product (; ). In [5] it is shown that C1 is equivalent to the condition that the smoothing iteration (3.3) converge at a rate exceeding that of a Richardson iteration dened by R =!?1 I with! = C?1 R. That paper then proves C1 for a class of smoothers dened by subspace decomposition and which satisfy simple hypotheses. In particular, point, line, and bloc Jacobi or Gauss-Seidel iterations satisfy C1. Remar 5. The multigrid algorithms described in [5, 8] use the following iteration as a smoothing x i = x i?1 + R (i+m()) (f? A x i?1 ) This iteration diers from (3.4) only notationally. In fact, the meaning given to R (e.g. the sweep direction for Gauss-Seidel smoothing) can be chosen, perhaps dierently, for each notation so that the two iterations are identical. 2 The second condition is expressed in terms of the adjoint P?1 W! W?1 of I taen with respect to the inner products A (; ) = (A ; ) and A?1 (; ), (3.10) A?1 (P?1 u; v) = A (u; I v); 8 v 2 W?1 C.2) \Approximation and Regularity" for some 2 (0; 1] with C independent of, 0 1 2 ja ((I? I P?1 )u; u)j C @ jjja ujjj2 0; (3.11) A A (u; u) 1? ; 8 u 2 W In Section 4.2 we will see that C2 is a consequence of the approximation properties of the spaces W and the elliptic regularity of the underlying partial dierential equation. Consider rst a result for the W-cycle Theorem 6. (W-cycle) If the conditions C1 and C2 are satised, then the m- smoothing W-cycle iteration dened by Algorithm 1 converges for suciently large m, and E = I? B A is a contraction, with contraction number (independent of ) given by M M + m ; where M is a constant, M = M(; C ; C R ). Furthermore, the same conclusion holds if \m is suciently large" is replaced by the assumption (3.12) A (I u; I u) 2 A?1 (u; u); 8 u 2 W?1 14

An explicit expression for M can be found in [6]. For the bilinear forms A (; ) obtained in the next section for the modied Ciarlet-Raviart method (2.11), the condition (3.12) fails. And as previously noted, \m suciently large" can mean m 8 in this context, see Section 6. Not only will the W-cycle diverge for small m, but numerical results show that for the W-cycle, the preconditioned operator B j A j may be indenite. It is not certain that a conjugate gradient iteration for (3.8) will then converge. The next Theorem will demonstrate that one does not encounter this diculty with the Variable V-cycle. Theorem 7. (Variable V-cycle) If the conditions C1 and C2 are satised, then the Variable V-cycle multigrid algorithm yields preconditioners B such that (3.13) 0 A (u; u) A (B A u; u) 1 A (u; u); 8 u 2 W ; with (3.14) 0 m() M + m() ; 1 Y i=1 1 + C m(i) M + m() m() ; where M is a constant, M = M(; C ; C R ). As consequence of this theorem, the spectral condition number of B A satises 2 (B A ) 1 0 M + m() m() 2 ; and the multigrid-preconditioned conjugate gradient solution of (3.8) converges with an asymptotic rate of (3.15) q 2 (B j A j )? 1 q 2 (B j A j ) + 1 M M + 2m(j) per iteration. In Section 4.2 a proof is given of C2 for the mixed methods of Section 2.1. The preconditioning properties of the simple V-cycle are questionable in comparison to those of the Variable V-cycle. Numerical experiments described in Section 6 suggest that this condition number may not be bounded independent of the mesh. 3.3. Implementation. It remains to interpret the multigrid discussion of the previous section in the context of the mixed methods of Section 2.1. We provide such an interpretation in the next paragraphs. In particular, we relate the operator equation (3.1) to the Schur complement formulations (2.11) and (2.8). With this relation established, we then consider the question of the computability of Algorithms 1 and 2 which were expressed in terms of operators. A description of a multigrid-preconditioned conjugate gradient algorithm is given to end the section. Consider a nested sequence of quasi-uniform (in the sense of Ciarlet [13]) triangulations of a domain with mesh diameters fh g =1;...;j satisfying the growth condition (3.16) h?1 h ; 15

with independent of. Beginning with a coarse triangulation h1 of, such a sequence may be obtained by joining the midpoints of the edges of mesh h?1 to form mesh h. On each mesh construct nite dimensional spaces fw h ; V h g and associated mixed methods as outlined in Section 2.1. In particular, denote by f d; ; u d; g 2 V h W h the solution of (2.10). (In the sequel, subscripts h will be replaced by the subscript. Clearly these spaces are nested, W?1 W and V?1 V.) Let (; ) denote an inner product for the space W. We shall additionally assume that the associated norms jjjujjj 0; = (u; u) 1=2 are uniformly (with respect to ) equivalent to the L 2 () norm. (The inner products (; ) are allowed to dier from the discrete inner products (; ) h of Section 5!) Given local bases f i g and f'i g for W and V one obtains the following matrix equation for the coecient vector U d; of u d; with u d; = P i Ud; i i, (3.17) M U d; = B d; A?1 d; Bt d; U d; = F ; (compare (2.11) with f = F ). Recall that the matrices [B d; ] ij =?D (' j ; i ), and [A d; ] ij = (' i ; 'j ) h are sparse and computable. (A d; may be assumed to be diagonal according to Section 5.) Similarly, the data vector [F ] i = (f; i ) is computable. To apply the multigrid algorithms, dene operators A W! W and associated bilinear forms A (; ) satisfying (3.18) (A u; v) A (u; v) hm U; V i; 8 u; v 2 W ; where U and V are the coecient vectors of u and v, and h; i denotes the Euclidean inner product. One may interpret M as the stiness matrix associated with A and the basis f i g. Dening an element f 2 W such that (f ; i ) = [F ] i, we have the following reinterpretation of (3.17), (A u d; ; i ) = [M U d; ] i = [F ] i = (f ; i ) ; or A u d; = f The symmetry and positivity of the form A (; ), and hence of the operator A, follows from that of M. It is also easy to prove that A is positive denite, cf. Lemma 9. Finally, observe that as a result of the embedding W?1 W, one may choose the \prolongation" operator I of the previous section to be the natural injection. Remar 8. In the multigrid analysis of the next sections it will be convenient to have the following representation (3.19) A (u; u) 1=2 = h B d; A?1 d; Bt d; U; Ui1=2 = ha?1 d; Bt d; U; A d;a?1 d; Bt d; Ui1=2 ha d; (A?1 d; = sup Bt d; U); V i v2v nf0g ha d; V ; V i 1=2 D (v; u) = sup 1=2 v2v nf0g (v; v) ; h maing use of the fact that ha d; ; i induces an inner product for the space V. 2 16

According to the preceding discussion, it is convenient to pass the function f j to the multigrid algorithms in the form of the computable cartesian vector F j with entries [F j ] i = (f j ; i j ) j. It is also natural to represent the output of these algorithms, zj l = Mg j(z l?1 j ; f j ) and y j = B j g = Mg j (0; g) in terms of coecient vectors. Extending the notation slightly, Zj l = Mg j(z l?1 j ; F j ). The computability of Algorithms 1 and 2 is then determined by three factors the computability of f?1 P?1 (f?a x m() ); or more precisely, that the vector with entries [F?1 ] i = (f?1 ; i?1 ) is computable,?1 the computability of the coecient vector Q of q = I q?1 from Q?1, and the computability of the smoothing iterations. An inductive argument will then show Z l j and Y j to be computable since Zl 1 = Y 1 = M?1 1 F 1. Denoting by X m() the coecient vector of x m(), one can describe ~ f f? A x m() by the vector ~ F with entries [ ~ F ] i ( ~ f ; i ) = (f? A x m() ; i ) = [F? M X m() ] i If the basis functions for each space W are locally dened, then a representation for f?1 = P?1 f ~ is easily computed from F ~ since [F?1 ] i = (f?1 ; i?1 )?1 = (P?1 ~ f ; i?1 )?1 = ( ~ f ; I i?1 ) and I i?1 is a nown linear combination of a small number of the basis functions for W. Similarly, the coecient vector of q = I q?1 is easily obtained from that of q?1. The implementation of a class of smoothing iterations of the form (3.4) is considered in [5], see Remar 5. In particular, from an initial guess X i?1, (the coecient vector of x i?1 ), one may compute the coecient vector X i from a point, line, or bloc Jacobi or Gauss-Seidel iteration applied to the stiness matrix. (For these smoothers, the operators R and A depend upon the choice of inner products for W. However, it is observed that the smoothing iterations (3.4) are themselves independent of the choice for (; ).) The implementation of a Richardson iteration, dened by R =!?1 I, requires an additional step. In this case, the smoothing iteration satises (x i? x i?1 ; i ) =!?1 (f? A x i?1 ; i ) for all i. In terms of the Gramm matrix [G ] ij = ( i ; j ), this becomes G (X i? X i?1 ) =!?1 (F? M X i?1 ) ; and G must therefore be inverted for each smoothing step. However, if one selects appropriate discrete inner products; for example, (u; v) P = h 2 i U iv i, then the Gramm matrices can be made to be diagonal. It is not in general desirable to apply Jacobi or Gauss-Seidel smoothers to the dense matrix B A?1 Bt. By contrast, when the matrix A d; is diagonal, the Schur complement M = B d; A?1 d; Bt d; is sparse with entries that can be inexpensively computed. Since A d; and A are spectrally equivalent, it can be determined from the representation (3.19) that B d; A?1 d; Bt d; and B A?1 Bt are also spectrally equivalent. One may then precondition B A?1 Bt with the multigrid operator B constructed for B d; A?1 d; Bt d;. As was noted in 17

in the Introduction, this preconditioner may then be coupled with the technique of [7] to solve (2.7) without inverting A. According to Theorem 7, if B j is dened by a Variable V-cycle iteration, the problem B j A j u d;j = B j f j may be rapidly solved with a conjugate gradient iteration. Noting that B j A j is symmetric in the inner product [; ] = (B?1 j ; ) j, such a conjugate gradient iteration, cf. [23], is given in Algorithm 3. l = 0; x 0 = 0; z 0 = B j f j while z l > z 0 l = l + 1 if l = 1 p 1 = z 0 else = [z l?1 ; z l?1 ] = [z l?2 ; z l?2 ] p l = z l?1 + p l?1 end = [z l?1 ; z l?1 ] = [B j A j p l ; p l ] x l = x l?1 + p l z l = z l?1? B j A j p l end Algorithm 3. A conjugate gradient algorithm for (3.8). In this algorithm, z l = B j f j? B j A j x l denotes the l th residual. It is never necessary to compute B j?1. In fact, if one introduces a new variable r l satisfying z l = B j r l, then [z l?1 ; z l?1 ] = (r l?1 ; z l?1 ) j and [B j A j p l ; p l ] = (A j p l ; p l ) j. If additionally, [R l ] i (r l ; i ) j, then for coecient vectors one has (r l?1 ; z l?1 ) j = (R l?1 ) t Z l?1, (A j p l ; p l ) j = (P l ) t M j P l, and R l = R l?1?m j P l. The conjugate gradient algorithm for (3.8) may then be rewritten as Algorithm 4. Algorithm 4 is identical to the preconditioned conjugate gradient algorithm l = 0; X 0 = 0; R 0 = F j while Z l > Z 0 compute Z l = Mg j (0; R l ) [z l = B j r l ] l = l + 1 if l = 1 P 1 = Z 0 else = (R l?1 ) t Z l?1 = (R l?2 ) t Z l?2 P l = Z l?1 + P l?1 end = (R l?1 ) t Z l?1 = (P l ) t M j P l X l = X l?1 + P l R l = R l?1? M j P l end Algorithm 4. A multigrid-preconditioned conjugate gradient algorithm. 18

for the problem B j 1=2 A j B j 1=2 (B j?1=2 u d;j ) = B j 1=2 f j described in [17]. 4. Approximation and Regularity. In the next two sections I shall interpret and prove the estimate (3.11). More precisely, a proof is given of (3.11) for the perturbed Ciarlet-Raviart method of Section 2.1. The proof employs several properties of the Ciarlet- Raviart mixed method but can be extended to other mixed methods. 4.1. A scale of discrete-space norms. We rst prove several technical Lemmas which will be used in the next section. It is convenient to interpret the technical Lemmas in terms of a scale of discrete norms generated from powers of the operator A. That fractional and negative powers of A are well-dened is a consequence of the rst Lemma 9. Assume that H1, H2, and H4 are satised. Then A dened by (3.17) and (3.18) is self-adjoint with respect to the inner product (; ) and positive denite. proof According to the denition (3.18) of A for the perturbed Ciarlet-Raviart method (2.10), (4.1) (A u; v) = h B d; A?1 d; Bt d;u; V i = ha?1 d; Bt U; Bt V i That A is self-adjoint and positive is then a consequence of the symmetry and positivity of the Gramm matrix A d;, compare H2. As a consequence of H1, H2, and H4, the perturbed Ciarlet-Raviart method (2.10) has a unique solution. If there exists a nontrivial w 2 W such that A w = 0, then setting u = v = w in (4.1) one obtains B t W = 0. Equivalently, D ('; w) = 0 for all ' 2 V. This contradicts the uniqueness proved for u d;h of (2.10). Let f i; g i=1;...;n and f j g j=1;...;n denote the eigenvalues and associated eigenvectors of A. Tae the N eigenvectors to be orthonormal with respect to the inner product (; ). One may then dene the following scale of norms on W, 2 3 X jjjujjj s; = jjja s=4 ujjj 0; = 4 N 1=2 (4.2) s=2 i; c2 5 i where u = P N i=1 c i i. We now observe that jjjjjj 2; associated with the norm for H 2 (). i=1 exhibits an inverse property typically Lemma 10. If is quasi-uniform (so that H3 is valid) and if H2 is satised, then (4.3) jjjujjj 2; C h?2 u 0 ; 8 u 2 W ; and = max (A ) C 2 h?4 ; with a constant C which is independent of h. 19

proof The bound (4.3) is a simple consequence of the representation (3.19), the coercivity of (; ) h, the boundedness of the form D (; ), and the \inverse property" H3 since D ('; u) ' 1 u 1 C h?1 ' 0 C h?1 u 0 ; 8 f'; ug 2 V W The eigenvalue estimate then follows from the observation (A u; u) C 2 h?4 (u; u) jjjujjj 2; = jjja 1=2 ujjj 0; = (A u; u) 1=2 ; and the equivalence of the norms 0 and jjjjjj 0; on W. It would be convenient for the proof of Approximation and Regularity to have the equivalence on W of jjjjjj 2; and the H 2 () norm. Of course, such an equivalence cannot be obtained as the space W is not contained in H 2 (). In order to circumvent this diculty we prove and later apply the following Lemma 11. Assume that H2 is satised and that is a quasi-uniform triangulation of (or assume H3). For s in [0; 2] with s 6= 1=2; 3=2, denote by Q the L 2 -projection of H s 0 () into W which is given the norm jjjjjj s;. Then Q is bounded, i.e. (4.4) jjjq jjj s; C s ; 8 2 H s 0() ; with C independent of h. proof Consider rst the case s = 2. According to (3.19), and the coercivity of (; ) h from H2, obtained (4.5) jjjq jjj 2; = A (Q ; Q ) 1=2?1=2 D (v; Q ) sup v2v nf0g v 0?1=2 sup v2v nf0g D (v; ) + D (v; Q? ) v 0 For 2 H0 2 (), a standard Green's formula gives (4.6) D (v; ) =?(v; 4) L2 v 0 2 Using an approximation property of Q and an inverse property (4.7) D (v; Q? ) v 1? Q 1 C 1 h?1 v 0 C 2 h 2 C 1 C 2 v 0 2 20

Combining (4.5), (4.6), and (4.7), we obtain (4.8) jjjq jjj 2; C 2 ; 8 2 L 2 () ; with C independent of h. This completes the proof for s = 2. Additionally, the uniform equivalence of jjjjjj 0; and 0 yields (4.9) jjjq jjj 0; C Q 0 = C 0 ; 8 2 H 2 0() The bound (4.4) is then obtained by interpolating (4.8) and (4.9) that is, by interpolating the operator Q in the sense of Peetre, cf. [11]. This interpolation argument assumes that the Peetre intermediate norms (denoted here by s; ) on W are equivalent to the norms jjjjjj s;, independent of h. The following verication of this equivalence is standard. Consider the Peetre K-functional K(u; t) 2 = inf v2w jjju? vjjj 2 0; + t2 jjjvjjj 2 2; Using the representations u = P N i=1 c i i and v = P N i=1 b i i this becomes X K(u; t) 2 N i = inf h(c i? b i ) 2 + t 2 i; bi 2 ; b i i=1 with the inmum being attained for the choice b i = (t 2 i; + 1)?1 c i so that K(u; t) 2 = XN i=1 Consequently, one nds for the Peetre norm i; t 2 (t 2 i; + 1)?1 c 2 i u 2 s; def = = = Z 1 0 XN ( i=1 N ( X i=1 K(u; t) 2 t?2( s 2 )?1 dt s 2 i; c 2 i s 2 i; c 2 i = c s jjjujjj 2 s; Z 1 0 Z 1 0 (t 2 i; ) 1?( s 2 ) dt t 2 i; + 1 t (t 2 ) 1?( s 2 ) t 2 + 1 dt t ) ) A contour integration gives c s = 2 s sin 2. Since the K-method yields intermediate spaces H0() s = [H0(); 2 L 2 ()] 1? s provided that s 6= 1=2; 3=2 for polygonal this is a consequence of the Stein extension operator the proof is complete. 2 To conclude this section we record a consequence of the Schwarz inequality for the discrete norms (4.2). In particular, one has (4.10) jjjujjj 2?1; jjjujjj?2; jjjujjj 0; 21

The Schwarz inequality for sums gives jjjujjj 2?1; = XN i=1 0 @ X N i=1? 1 2 i; c2 i = 2(? 1 2) i; c 2 i XN i=1? 1 2 i; (c2 i ) 1 2 1 0 1 2 A @ X N c 2 i i=1 1 A 1 2 h i (c 2 i ) 1 2 = jjjujjj?2; jjjujjj 0; 4.2. Proof of (3.11). Assume that the forms (; ) h have been chosen so that bounds of type (2.18) and (2.19) are satised for an appropriate value of s. Since, according to the Schwarz inequality, A ((I? I P?1 )u; u) A ((I? I P?1 )u; (I? I P?1 )u) 1=2 A (u; u) 1=2 ; it suces to show that (4.11) A ((I? I P?1 )u; (I? I P?1 )u) 1=2 C? jjja ujjj 2 0; A (u; u) 1 2? ; with to be determined. Essential to the method of proof is the identication of (I? I P?1 )u as an error in the following sense. One may construct an f 2 H?1 () and an associated system (2.10) for which u (an element of W ) and I P?1 u are solutions associated with meshes and?1, provided that I is the natural injection. To this end, for any u 2 W, consider the element G u 2 W satisfying (4.12) (G u ; ) L2 = A (u; ) ; 8 2 W Since H0 1() is continuously embedded in L 2(), (G u ; ) L2 denes an element of H?1 (), which we also denote by G u. Consider then the continuous problem (2.5), and the problem (2.10) for the two meshes?1 and, all with data f =?G u. Denote the respective solutions of these problems by f; wg 2 H 1 () H0(), 1 and f d;l ; w d;l g 2 V l W l, for l =? 1;. The equation for the associated Schur complement on the th mesh, in the notation of Section 3.3, is A (w d; ; ) = (G u ; ) L2 ; 8 2 W Therefore u = w d; by construction. Similarly, on the (? 1) st mesh A?1 (w d;?1 ; ) = (G u ; ) L2 ; 8 2 W?1 But according to the denition of P?1, and if I was taen to be the natural injection operator A?1 (P?1 u; ) = A (u; I ) = (G u ; I ) L2 = (G u ; ) L2 ; 8 2 W?1 Since A?1 (; ) is an inner product for W?1, we conclude that I P?1 u = w d;?1. Having interpreted the term (I? I P?1 )u, one may apply approximation properties for f d;l ; w d;l g for l =? 1;, and elliptic regularity to prove 22

Lemma 12. If (; ) h is chosen so that the estimates (2.18) and (2.19) are valid with s = 3, if H2 is satised, and if the nested meshes are quasi-uniform (H3) and satisfy (3.16), then (4.13) A ((I? I P?1 )u; (I? I P?1 )u) 1=2 C?1=4 G u?1 with C independent of h. proof Beginning with the identication provided above, and by (3.19) and H2, (4.14) A ((I? I P?1 )u; (I? I P?1 )u) 1=2 = A (w d;? w d;?1 ; w d;? w d;?1 ) 1=2?1=2 D (v; w d;? w d;?1 ) sup v2v nf0g v 0 The estimates (2.17), and (2.19) with s = 3, H3, and the boundedness of D (; ), combine to give (4.15) D (v; w d;? w d;?1 ) C w? w d; 1 + w? w d;?1 1 v 1 C h 2 + h 2?1 w 3 Ch?1 v 0 Ch w 3 v 0 since h?1 h, see (3.16). Applying the regularity bound, w 3 C G u?1 ; the proof is completed with the observation that h C?1=4, cf. Lemma 10. Comparing (4.11) and (4.13), a natural choice for is 1 4. Dening G u; = A u so that (G u; ; ) = (A u; ) = A (u; ) = (G u ; ) L2 ; 8 2 W ; then, as a consequence of Lemma 11, (4.16) G u?1 C jjjg u; jjj?1; Indeed, if Q denotes the L 2 projection onto W, G u?1 = and from the Schwarz inequality, sup 2H 1 0 () (G u ; ) L2 1 C sup 2H 1 0 () (G u ; ) L2 jjjq jjj 1; ; (G u ; ) L2 = (G u ; Q ) L2 = (G u; ; Q ) jjjg u; jjj?1; jjjq jjj 1; 23

It remains then to prove that jjjg u; jjj?1; C jjja ujjj 0; 1 2 A (u; u) 1 2 ( 1 2 ) This inequality is an immediate consequence of the inequality (4.10) and the identications and jjjg u; jjj?1; jjjg u; jjj 1=2 0; jjjg u;jjj 1=2?2; ; jjjg u; jjj 0; = jjja ujjj 0; ; jjjg u; jjj?2; = jjja?1=2 G u; jjj 0; = jjja 1=2 ujjj 0; = (A u; u) 1=2 = A (u; u) 1=2 5. Approximate Forms. As discussed in Section 2.1, the motivation for introducing approximate forms (; ) h was a desire to avoid inverting the Gramm matrix [A ] ij = (' i ; ' j ) L2. It was observed in Section 2.3 that approximating (; ) L2 in (2.7) with (; ) h to obtain (2.10) led to additional terms in the error estimates, but that these terms are conveniently controlled if (5.1) and (5.2) je ( ; v)j sup Ch s?2 u v2v nf0g v s ; 0 E ( d; ; y d ) sup Ch s?1 u d2h?1 ()nf0g d s?1 It was also necessary to assume the degree of the piecewise polynomial spaces V and W to be greater than or equal to s? 1. In this section we will describe forms (; ) h and local bases f ~' i g for the quadratic and cubic Ciarlet-Raviart spaces V, with respect to which the new Gramm matrix A d; is diagonal, [A d; ] ij = ( ~' i ; ~' j ) h. The relations (5.1) and (5.2), and the assumption H2 will then be proved for these forms. The approximate forms (; ) h will be constructed locally from certain quadrature schemes dened on triangles T of the mesh, X Z (5.3) (u; v) h = (u; v) h ;T ; with (u; v) h ;T Q T (u; v) uv dx T 2 T Corresponding to the local forms we have the local errors Z X (5.4) E T (u; v) = (u; v) h ;T? uv dx ; with E (u; v) = E T (u; v) T T 2 It will be possible to prove (5.1) and (5.2) provided that the local quadrature schemes are suciently accurate. If, for example, on each triangle T (5.5) je T (u; v)j C h 2 T u 1;T v 1;T ; 8 u; v 2 P (2) (T ) ; 24