JOHANNES KEPLER UNIVERSITY LINZ. A Decomposition Result for Biharmonic Problems and the Hellan-Herrmann-Johnson Method

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1 JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics A Decomposition Result for Biharmonic Problems and the Hellan-Herrmann-Johnson Method Wolfgang Krendl Doctoral Program Computational Mathematics (W1214) Johannes Kepler University Altenbergerstraße 69, 4040 Linz, Austria Katharina Rafetseder Institute of Computational Mathematics, Johannes Kepler University Altenberger Str. 69, 4040 Linz, Austria Walter Zulehner Institute of Computational Mathematics, Johannes Kepler University Altenberger Str. 69, 4040 Linz, Austria NuMa-Report No July 2014 Rev. 1 April 2016 A 4040 LINZ, Altenbergerstraße 69, Austria

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3 A DECOMPOSITION RESULT FOR BIHARMONIC PROBLEMS AND THE HELLAN-HERRMANN-JOHNSON METHOD WOLFGANG KRENDL, KATHARINA RAFETSEDER, AND WALTER ZULEHNER Abstract. For the first biharmonic problem a mixed variational formulation is introduced which is equivalent to a standard primal variational formulation on arbitrary polygonal domains. Based on a Helmholtz decomposition for an involved nonstandard Sobolev space it is shown that the biharmonic problem is equivalent to three (consecutively to solve) second-order elliptic problems. Two of them are Poisson problems, the remaining one is a planar linear elasticity problem with Poisson ratio 0. The Hellan-Herrmann-Johnson mixed method and a modified version are discussed within this framework. The unique feature of the proposed solution algorithms for the Hellan-Herrmann- Johnson method and its modified variant is that they are solely based on standard Lagrangian finite element spaces and standard multigrid methods for second-order elliptic problems and they are of optimal complexity. Key words. biharmonic equation, Hellan-Herrmann-Johnson method, mixed methods, Helmholtz decomposition AMS subject classifications. 65N22, 65F10, 65N55 1. Introduction. We consider the first biharmonic boundary value problem: For given f, findw such that (1.1) 2 w = f in, w = n w = 0 onγ, where is an open and bounded set in R 2 with a polygonal Lipschitz boundary Γ, and n denote the Laplace operator and the derivative in the direction normal to the boundary, respectively. Problems of this type occur, for example, in fluid mechanics, where w is the stream function of a two-dimensional Stokes flow, see, e.g., [22], and in linear elasticity, wherew is the vertical deflection of a clamped Kirchhoff plate, see, e.g., [18]. In this paper we focus on finite element methods for discretizing the continuous problem (1.1). The aim is to construct and analyze efficient iterative methods for solving the resulting linear system. In particular, the Hellan-Herrmann-Johnson (HHJ) mixed finite element method is studied, see [26, 27, 30], which is strongly related to the non-conforming Morley finite element, see [33, 2]. The proposed iterative method consists of the application of the preconditioned conjugate gradient method to three discretized elliptic problems of second order. The implementation requires only manipulations with standard conforming Lagrangian finite elements for second-order problems. The proposed preconditioners are standard multigrid preconditioners for second-order problems, which lead to mesh-independent convergence rates. The results are based on a decomposition of the continuous problem into three secondorder elliptic problems, which are to be solved consecutively. The first and the last problem are Poisson problems with Dirichlet conditions, the second problem is a pure traction problem in planar linear elasticity with Poisson ratio 0. The HHJ method is a non-conforming method in this setup. A conforming modification will be discussed as well. There are many alternative approaches for biharmonic problems discussed in literature. Finite element discretizations range from conforming and classical non-conforming finite element methods for fourth-order problems, discontinuous Galerkin (dg) methods for The research was supported by the Austrian Science Fund (FWF): W1214-N15, project DK12 and S11702-N23. Doctoral Program Computational Mathematics, Johannes Kepler University Linz, 4040 Linz, Austria (wolfgang.krendl@dk-compmath.jku.at). Institute of Computational Mathematics, Johannes Kepler University Linz, 4040 Linz, Austria (rafetseder@numa.uni-linz.ac.at,zulehner@numa.uni-linz.ac.at). 1

4 2 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER fourth-order problems to various mixed methods including mixed dg methods, see, e.g., [18, 20, 15, 5, 7, 4, 29], and the references cited there. Solution techniques proposed for the linear systems, which show mesh-independent or nearly mesh-independent convergence rates are typically based on two-level or multilevel additive or multiplicative Schwarz methods (including multigrid methods), see, e.g., [36, 12, 13, 38, 14, 39, 16, 25], and the references cited there. We are not aware of any other approach, which is based solely on standard components for second-order elliptic problems (regarding both the discretization and the solver for the discretized problem) and for which optimal convergence behavior could be shown. Regarding the discretization, the method introduced in [5, 7] also uses only standard C 0 finite element spaces for second-order problems, however for a different formulation in the kinematic variables w and w. The analysis of the method in [5, 7] is based on a mesh-dependent energy norm, which requires some extra nonstandard techniques for constructing and analyzing an efficient iterative solver. The Hellan-Herrmann-Johnson method is strongly related to the mixed dg method introduced in [29] for plate bending problems. This method allows a reduction to a linear system for an approximation ofw from a standardc 0 finite element space, where the associated system matrix can be seen as a discretization matrix of a fourth-order operator (in our case of the biharmonic operator). This again requires some extra techniques beyond the case of second-order problems for constructing and analyzing an efficient iterative solver. An additional feature of the approach in this paper is a new formulation of the underlying continuous mixed variational problem, which is fully equivalent to the original primal variational problem without any further assumptions on like convexity. This is achieved by introducing an appropriate nonstandard Sobolev space. The results of this paper can be extended to boundary conditions of the formw = w = 0 on Γ, which represent simply supported Kirchhoff plates in linear elasticity, as well as to non-homogeneous variants. An extension to plate bending problems with free boundaries and, more generally, to mixed variants involving all three types of boundary conditions is not straight-forward and subject of further investigations. The paper is organized as follows. Section 2 contains a modification of a standard mixed formulation of the biharmonic problem, for which well-posedness will be shown. A Helmholtz decomposition of an involved nonstandard Sobolev space is derived in Section 3 and the resulting decomposition of the biharmonic problem is presented. In Section 4 the Hellan-Herrmann-Johnson method and a modified version are discussed. Section 5 contains the discrete version of the Helmholtz decomposition of Section 3. In Section 6 we briefly discuss error estimates. The paper closes with a few numerical experiments in Section 7 for illustrating the theoretical results. 2. A modified mixed variational formulation. Here and throughout the paper we use L 2 (),W m,p (), H m () = W m,2 (), andh m 0 () with its dual spaceh m () to denote the standard Lebesgue and Sobolev spaces with corresponding norms. 0,. m,p,. m =. m,2,. m, and. m forp 1 and positive integersm, see, e.g., [1]. A standard (primal) variational formulation of (1.1) reads as follows: For given f H 1 (), findw H 2 0() such that (2.1) 2 w : 2 v dx = f,v for all v H 2 0(), where 2 denotes the Hessian, A : B = 2 i,j=1 A ij B ij for A,B R 2 2, and, denotes the duality product in H H for a Hilbert space H with dual H, here for H =

5 THE HELLAN-HERRMANN-JOHNSON METHOD 3 H0 1 (). Existence and uniqueness of a solution to (2.1) is guaranteed even for more general right hand sidesf H 2 () by the theorem of Lax-Milgram, see, e.g., [34, 32]. For the HHJ mixed method the auxiliary variable (2.2) w = 2 w is introduced, whose elements can be interpreted as bending moments in the context of linear elasticity. This allows to rewrite the biharmonic equation in (1.1) as a system of two secondorder equations (2.3) 2 w = w, divdivw = f in with the following notations for a matrix-valued function v and a vector-valued function φ. [ ] 1 v divv = v 12 and divφ = 1 v v 1 φ φ In the standard approach a mixed variational formulation is directly derived from the system (2.3). We take here a little detour, which better motivates the nonstandard Sobolev space we use in this paper for a modified mixed variational formulation. Starting point is the following unconstrained optimization problem: Findw H0 2 () such that (2.4) J(w) = min v H 2 0 () J(v) with J(v) = 1 2 v : 2 v dx f,v. 2 It is well-known that (2.4) is equivalent to (2.1). Actually, (2.1) can be seen as the optimality system characterizing the solution of (2.4). By introducing the auxiliary variablew = 2 w L 2 () sym with L 2 () sym = {v: v ji = v ij L 2 (), 1 i,j 2}, equipped with the standardl 2 -norm v 0 for matrix-valued functionsv, the objective functional becomes a functional depending on the original and the auxiliary variable: (2.5) J(v,v) = 1 v : v dx f,v. 2 The weak formulation of (2.2) leads to the constraint (2.6) c((w,w),τ) = 0 for all τ M, where c((v,v),τ) = v : τ dx v divτ dx, and M is a (not yet specified) space of sufficiently smooth matrix-valued test functions. By this the unconstrained optimization problem (2.4) is transformed to the following constrained

6 4 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER optimization problem: Find(w,w) H 1 0 () L2 () sym that minimizes the objective functional (2.5) subject to the constraint (2.6). The Lagrangian functional associated with this constrained optimization problem is given by L((v,v),τ) = J(v,v)+c((v,v),τ), which leads to the following first-order optimality system: w : v dx+c((v,v),σ) = f,v for all (v,v) H0 1 (2.7) () L2 () sym, c((w,w),τ) = 0 for all τ M. Here σ M denotes the Lagrangian multiplier associated with the constraint (2.6). The optimality system is a saddle point problem on the spacex = H0() L 1 2 () sym, equipped with the standard norm (v,v) X = ( v 2 ) 1/2 1 + v 2 0 for the primal variable (v,v) and the (not yet specified) Hilbert space M, equipped with a norm τ M for the dual variable τ. An essential condition for the analysis of (2.7) is the inf-sup condition for the bilinear form c, which reads: There is a constant β > 0 such that It is easy to see that c((v,v),τ) sup β τ M, 0 (v,v) X (v,v) X c((v,v),τ) (2.8) sup = ( τ divdivτ 2 ) 1/2 1 0 (v,v) X (v,v) X for sufficiently smooth functions τ. If the right-hand side in (2.8) is chosen as the norm in M, then the inf-sup condition is trivially satisfied with constant β = 1. This motivates to set M = H 1 (divdiv,) sym with equipped with the norm (2.9) H 1 (divdiv,) sym = {τ L 2 () sym : divdivτ H 1 ()}, τ 1,divdiv = ( τ divdivτ 2 1) 1/2. Heredivdivτ is meant in the distributional sense. It is easy to see thath 1 (divdiv,) sym is a Hilbert space. In order to have a well-defined bilinear form c, the original definition has to be replaced by c((v,v),τ) = v : τ dx+ divdivτ,v, which coincides with the original definition, if τ is sufficiently smooth, say τ H 1 () sym with H 1 () sym = {τ L 2 () sym : τ ij H 1 (), 1 i,j 2}, equipped with the standard H 1 -norm τ 1 and H 1 -semi-norm τ 1 for matrix-valued functionsτ. Observe that H 1 () sym H 1 (divdiv,) sym L 2 () sym.

7 THE HELLAN-HERRMANN-JOHNSON METHOD 5 From the first row of the optimality system (2.7) forv = 0 it easily follows that w = σ. So the auxiliary variable w can be eliminated and we obtain after reordering the following reduced optimality system: For f H 1 (), find σ H 1 (divdiv,) sym and w H0 1 () such that σ : τ dx divdivτ,w = 0 for all τ H 1 (divdiv,) sym, (2.10) divdivσ,v = f,v for all v H0(). 1 REMARK 2.1. The presented approach to derive the mixed method via the optimality system of a constrained optimization problem is the same approach as taken in [19] for the Ciarlet-Raviart mixed method. See [40] for a reformulation involving a similar nonstandard Sobolev spaceh 1 (,) = {v H 1 () : v H 1 ()} as in this paper. Problem (2.10) has the typical structure of a saddle point problem (2.11) a(σ,τ)+b(τ,w) = 0 for all τ V, b(σ,v) = f,v for all v Q. If the linear operatora: V Q (V Q) is introduced by [ [ σ τ A, = a(σ,τ)+b(τ,w)+b(σ,v), w] v] the mixed variational problem (2.11) can be rewritten as a linear operator equation [ [ σ 0 A =. w] f] If the bilinear forma is symmetric, i.e.,a(σ,τ) = a(τ,σ), and non-negative, i.e.,a(τ,τ) 0, which is the case for (2.10), it is well-known that A is an isomorphism from V Q onto (V Q), if and only if the following conditions are satisfied, see, e.g., [10]: 1. a is bounded: There is a constant a > 0 such that a(σ,τ) a σ V τ V for allσ, τ V. 2. b is bounded: There is a constant b > 0 such that b(τ,v) b τ V v Q for all τ V, v Q. 3. a is coercive on the kernel ofb: There is a constantα > 0 such that a(τ,τ) α τ 2 V for all τ kerb withkerb = {τ V : b(τ,v) = 0 for all v Q}. 4. b satisfies the inf-sup condition: There is a constant β > 0 such that inf sup 0 v Q0 τ V b(τ,v) τ V v Q β. Here τ V and v Q denote the norms in V and Q, respectively. We will refer to these conditions as Brezzi s conditions with constants a, b, α, and β. (We tacitly assume that a and b are the smallest constants for estimating the bilinear forms a and b. Then a and b match the standard notation for the norms of the bilinear forms a and b.)

8 6 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER In the next theorem we show that Brezzi s conditions are satisfied for (2.10). For the proof as well as for later use, we first introduce the following simple but useful notation for a functionv H0 1(): [ ] 1 0 (2.12) π(v) = vi with I =. 0 1 THEOREM 2.2. The bilinear forms a(σ,τ) = σ : τ dx and b(τ,v) = divdivτ,v satisfy Brezzi s conditions onv = H 1 (divdiv,) sym andq = H0 1 (), equipped with the norms τ 1,divdiv and v 1, respectively, with the constants a = b = α = 1 and β = (1+2c 2 F ) 1/2, wherec F denotes the constant in Friedrichs inequality: v 0 c F v 1 for allv H 1 0 (). Proof. 1. Letσ,τ H 1 (divdiv,) sym. Then a(σ,τ) σ 0 τ 0 σ 1,divdiv τ 1,divdiv. 2. Letτ H 1 (divdiv,) sym andv H0 1 (). Then b(τ,v) = divdivτ,v divdivτ 1 v 1 τ 1,divdiv v Observe thatkerb = {τ L 2 () sym : divdivτ = 0}. Therefore, a(τ,τ) = τ 2 0 = τ 2 1,divdiv for all τ kerb. 4. Here we follow the proofs in [17, 10]. Forv H0 1 () it is easy to see that b(π(v),v) = v 2 1 and π(v) 2 1,divdiv = π(v) v 2 1 (1+2c2 F ) v 2 1. Therefore sup 0 τ V b(τ,v) b(π(v),v) = τ 1,divdiv π(v) 1,divdiv 1 (1+2c 2 F )1/2 v 1. v 2 1 π(v) v 2 1 )1/2 COROLLARY 2.3. The problems (2.1) and (2.10) are fully equivalent, i.e., ifw H0() 2 solves (2.1), then σ = 2 w H 1 (divdiv,) sym and (σ,w) solves (2.10). And, vice versa, if(σ,w) H 1 (divdiv,) sym H0 1() solves (2.1), thenw H2 0 () andw solves (2.1). Proof. Both problems are uniquely solvable. Therefore, it suffices to show that (w, σ) with σ = 2 w solves (2.10), if w solves (2.1). So, assume that w H0 2 () is a solution of (2.1). Then, obviously,σ L 2 () sym and σ : 2 v dx = f,v for all v H0(), 2

9 THE HELLAN-HERRMANN-JOHNSON METHOD 7 which implies that divdivσ = f H 1 () in the distributional sense. Therefore, σ H 1 (divdiv,) sym and the second row in (2.10) immediately follows. By the definition ofdivdivτ in the distributional sense we have divdivτ,v = τ : 2 vdx v C0 (). SinceC 0 () is dense inh 2 0(), it follows forv = w that divdivτ,w = τ : 2 w dx = τ : σ dx, which shows the first row in (2.10). REMARK 2.4. The spaceh 1 (divdiv,) sym was already introduced in [37, 35] within linear elasticity problems - not for the plate bending problem of linear elasticity considered here, however. There is a natural trace operator associated with H 1 (divdiv,) sym, which was discussed in [37, 35]. We briefly recall here the basic properties for later reference. Let the boundaryγ of the polygonal domainbe written in the form Γ = K Γ k, k=1 whereγ k,k = 1,2,...,K, are the edges ofγ, considered as open line segments. Γ k denotes the corresponding closed line segment. Forτ H 1 (divdiv,) sym which are additionally twice continuously differentiable and v H 2 () H0() 1 we obtain the following identity by integration by parts. (2.13) (divdivτ)v dx = τ : 2 v dx τ nn n v ds Γ with the outer normal unit vectornofγand τ nn = n T τn. Following standard procedures this identity allows to extend the trace τ nn to all functions τ H 1 (divdiv,) sym as an element of the dual of the image of the Neumann traces of functions fromh 2 () H0 1 (), i.e. τ nn H 1/2 pw (Γ) = ΠK k=1 H 1/2 (Γ k ), whereh 1/2 (Γ k ) is the dual of H 1/2 (Γ k ), see [23] for details. Another widely used notation for H 1/2 (Γ k ) ish 1/2 00 (Γ k), see [32]. From (2.13) we obtain the corresponding Green s formula forτ H 1 (divdiv,) sym andv H 2 () H0 1(): (2.14) divdivτ,v = τ : 2 v dx τ nn, n v Γ. Here, Γ denotes the duality product in a Hilbert space of functions onγ.

10 8 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER 3. A Helmholtz decomposition ofh 1 (divdiv,) sym. In this section we study some important structural properties of H 1 (divdiv,) sym, which are helpful for analyzing the HHJ method in the next sections. THEOREM 3.1. For eachτ H 1 (divdiv,) sym, there is a unique decomposition τ = τ 0 +τ 1, where τ 0 = π(p) for some p H0 1() andτ 1 L 2 () sym with divdivτ 1 = 0. Moreover, c ( p τ 1 2 0) τ 2 1,divdiv c ( p τ 1 2 ) 0 for all τ H 1 (divdiv,) sym, with positive constants c and c which depend only on the constantc F of Friedrichs inequality. Proof. For τ H 1 (divdiv,) sym, let p H0 1 () be the unique solution to the variational problem (3.1) p v dx = divdivτ,v for allv H0 1 () and set τ 0 = π(p). Since divdivτ 0,v = p v dx, it follows thatdivdivτ 0 = divdivτ, and, therefore,divdivτ 1 = 0 forτ 1 = τ τ 0 in the distributional sense. On the other hand, if τ = τ 0 +τ 1 withτ 0 = π(p) anddivdivτ 1 = 0, then divdivτ 0 = divdivτ + divdivτ 1 = divdivτ, which implies (3.1). This shows the uniqueness. Furthermore, (3.1) implies τ = 2 p 2 1 = 2 divdivτ 2 1. Hence and τ 2 1,divdiv = τ divdivτ 2 1 = τ 0 +τ p τ τ p 2 1 ( 1+4c 2 F) p τ p τ = p τ τ p τ τ τ 2 0 +(1+4c2 F ) p 2 1 = 2 τ 2 0 +(1+4c2 F ) divdivτ 2 1. Then the estimates immediately follow with the constants1/c = c = max(2,1+4c 2 F ). In short, we have algebraically as well as topologically with H 1 (divdiv,) sym = π(h0 1 ()) H (divdiv,) H (divdiv,) = {τ L 2 () sym : divdivτ = 0}. Here denotes the direct sum of Hilbert spaces. REMARK 3.2. The Helmholtz decomposition of L 2 () sym in [28], based on previous results in [6], has the same second component. The first component in [6, 28] is different and requires the solution of a biharmonic problem in contrast to Theorem 3.1, where the first component requires to solve only a Poisson problem.

11 THE HELLAN-HERRMANN-JOHNSON METHOD 9 Next an explicit characterization of H (div div, ) is given. THEOREM 3.3. Let be simply connected. For each τ H (divdiv,), there is a functionφ ( H 1 () ) 2 such that τ = H T ε(φ)h with H = [ ] and ε(φ) ij = 1 2 ( jφ i + i φ j ). And vice versa, each function of the form τ = H T ε(φ)h with φ ( H 1 () ) 2 lies in H (divdiv,). The functionφis unique up to an element from RM = { τ(x) = a and there is a constantc K such that [ ] } x2 +b: a R, b R x 2, 1 c K φ 1 τ 0 = ε(φ) 0 φ 1 for allφ ( H 1 () ) 2 /RM. Proof. From Lemma 1 in [6] it follows that τ H (divdiv,) can be written in the following way: [ ] [ ] 0 ρ 2 φ τ = +Curlφ with Curlφ = 1 1 φ 1 ρ 0 2 φ 2 1 φ 2 for someρ L 2 0 () andφ ( H 1 () ) 2. The proof in [6] relies on the characterization of the kernel of the divergence operator in ( L 2 () ) 2, which is well-known (see, e.g., [22, Theorem 3.1]) and in ( H 1 () ) 2, for which we refer to Lemma 7.3 in the appendix. To ensure symmetry ofτ, it follows (see [28]) that ρ = 1 2 divφ. Replacing φ = (φ 1,φ 2 ) T by ( φ 2,φ 1 ) T yields the representation. The estimates follow from Korn s inequality. Therefore, we have the following representation of the solutionσ to (2.10): (3.2) σ = π(p)+h T ε(φ)h The analogous representation for the test functionsτ = π(q)+h T ε(ψ)h leads to following equivalent formulation of (2.10). Find p H0 1(), φ ( H 1 () ) 2 /RM, w H 1 0 () such that (3.3) π(p) : π(q) dx+ π(p) : ε(ψ) dx+ p v dx for all q H 1 0 (), ψ ( H 1 () ) 2 /RM,v H 1 0 (). π(q) : ε(φ) dx+ w q dx = 0, ε(φ) : ε(ψ) dx = 0 = f,v

12 10 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER Observe that π(p) : π(q) = 2pq and π(q) : ε(ψ) = q divψ, which allows to simplify parts of (3.3). In summary, the biharmonic problem is equivalent to three (consecutively to solve) elliptic second-order problems. The first problem is a Poisson problem with Dirichlet boundary conditions for p, which reads in strong form p = f in, p = 0 onγ. The second problem is a pure traction problem in linear elasticity with Poisson ratio 0 for φ, which reads in strong form divε(φ) = p in, ε(φ)n = 0 onγ. And, finally, the third problem is a Poisson problem with Dirichlet boundary conditions for the original variable w, which reads in strong form w = 2p+divφ in, w = 0 onγ. 4. The Hellan-Herrmann-Johnson method. Let T h be an admissible triangulation of the polygonal domain. Fork N the standard finite element spacess h ands h,0 are given by S h = {v C(): v T P k for all T T h } and S h,0 = S h H 1 0 (), wherep k denotes the set of bivariate polynomials of total degree less than or equal tok. A quite natural discretization of (3.3) is the following conforming method. Find p h S h,0, φ h (S h ) 2 /RM, w h S h,0 such that π(p h ) : π(q)dx + π(q) : ε(φ h ) dx+ w h q dx = 0, (4.1) π(p h ) : ε(ψ) dx+ ε(φ h ) : ε(ψ) dx = 0 p h v dx = f,v for all q S h,0, ψ (S h ) 2 /RM, v S h,0. In this and the subsequent section we will see that this method is strongly related to the HHJ method, which we introduce next. For the approximation of the Lagrangian multiplier σ, the HHJ method uses the finite element space V h = {τ L 2 () sym : τ T P k 1 for all T T h, and τ nn is continuous across inter-element boundaries}. For the approximation of the original variablew the standard finite element space Q h = S h,0 is used. So, the HHJ method reads as follows: Findσ h V h andw h Q h such that σ h : τ dx divdiv h τ,w h =0 for allτ V h, (4.2) divdiv h σ h,v = f,v for allv Q h

13 THE HELLAN-HERRMANN-JOHNSON METHOD 11 with divdiv h τ,v = T { T } τ : 2 v dx τ nn n v ds T forτ V h, v Q h. Similar to the linear functional divdivτ H 1 () (for τ H 1 (divdiv,) sym ), we consider divdiv h τ (for τ V h ) as a linear functional from the dual of Q h. And as introduced in Section 2 we use.,. as the generic symbol for duality products, here for the Hilbert spaceq h. If compared with (2.14), this definition of divdiv h τ,v forτ V h andv Q h in the HHJ method is just an element-wise assembled version of corresponding expression on the continuous level, a standard technique in non-conforming methods. REMARK 4.1. Using integration by parts we obtain divdiv h τ,v = T T h { T } divτ v dx τ ns s v ds T with the normal vectorn = (n 1,n 2 ) T, the vectors = ( n 2,n 1 ) T, which is tangent toγ, the tangential derivative s, and τ ns = s T τn. The HHJ method is often formulated with this representation, which allows an extension for all functions τ from the (mesh-dependent) infinite dimensional function space (4.3) Ṽ = {τ L 2 () sym : τ ij T H 1 (T) for allt T h, 1 i,j 2, and τ nn is continuous across inter-element boundaries}. This space was used for the analysis of the method in [17, 3, 21], and others. Existence and uniqueness of a solution for the corresponding variational problem on the continuous level could be shown under additional smoothness assumptions. For the approach taken in this paper, this is not required. Similar to the continuous case, the well-posedness of the discrete problem can be shown. For the proof of the discrete inf-sup condition the discrete analogue to π(v), see (2.12), is needed. Forv h S h,0, we define π h (v h ) = Π h π(v h ) with the linear operatorπ h, introduced in [17] by the conditions (4.4) ((τ h ) nn τ nn )q ds = 0, for all q P k 1, for all edgeseoft, T T h, e and (4.5) T ((τ h ) ij τ ij )q dx = 0, for all q P k 2, T T h, 1 i,j 2, for τ h = Π h τ V h and τ π(q h ). Observe that Π h was originally introduced in [17] as a linear operator on the infinite dimensional spaceṽ from above.

14 12 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER From the corresponding properties of Π h in [17], Lemma 4, the next result directly follows. LEMMA 4.2. Assume that T h is a regular family of triangulation. Then there exists a constantc B > 0 which is independent ofhsuch that π h (v) 0 c B v 1 for allv S h,0. Moreover, we need the following simple identity. LEMMA 4.3. For all p,v S h,0, we have divdiv h π h (p),v = p vdx. Proof. By integration by parts we have divdiv h π h (p),v = { Π h π(p) : 2 v dx T T h = T T h = T T h { { T T T T } (Π h π(p)) nn n v ds π(p) : 2 v dx (π(p)) nn n v ds T } p v dx p n v ds T } = p vdx. Now the well-posedness of the discrete problem can be shown. THEOREM 4.4. The bilinear forms a(σ,τ) = σ : τ dx, b h (τ,v) = divdiv h τ,v satisfy Brezzi s conditions onv h andq h, equipped with the norms τ 1,divdiv,h and v 1, respectively, where (4.6) and τ 1,divdiv,h = ( τ divdiv hτ 2 ) 1/2 1,h l 1,h = sup v h S h,0 l,v h v h 1 forl (S h,0 ), with the constants a = b = α = 1 and β = (1+c 2 B) 1/2, where c B denotes the constant in Lemma 4.2. Proof. 1. Letσ,τ V h. Then a(σ,τ) σ 0 τ 0 σ 1,divdiv,h τ 1,divdiv,h. 2. Letτ V h andv Q h. Then b(τ,v) = divdiv h τ,v divdiv h τ 1,h v 1 τ 1,divdiv,h v 1.

15 THE HELLAN-HERRMANN-JOHNSON METHOD Observe thatkerb h = {τ V h : divdivτ h = 0}. Therefore, a(τ,τ) = τ 2 0 = τ 2 1,divdiv,h forτ kerb h. 4. From Lemma 4.2 and Lemma 4.3 we obtain forv Q h and Therefore, sup 0 τ V h b h (π h (v),v) = v 2 1 π h (v) 2 1,divdiv,h = π h(v) v 2 1 (1+c2 B ) v 2 1. b h (τ,v) τ 1,divdiv,h b h(π h (v),v) π h (v) 1,divdiv,h = 1 (1+c 2 B )1/2 v 1. v 2 1 ( π h (v) v 2 1 )1/2 Observe that the norms introduced for the space V = H 1 (divdiv,) sym in (2.9) and its discrete counterpartv h in (4.6) are similar but different. For the discrete problem the norm is mesh-dependent. 5. A discrete Helmholtz decomposition. We have the following discrete version of Theorem 3.1. THEOREM 5.1. For eachτ V h, there is a unique decomposition τ = ˆτ 0 + ˆτ 1, where ˆτ 0 = π h (ˆp) for some ˆp Q h and ˆτ 1 V h with divdiv h ˆτ 1 = 0. Moreover, c ( ˆp ˆτ 1 2 0) τ 2 1,divdiv,h c ( ˆp ˆτ 1 2 ) 0 for all τ V h, with positive constantscand c, which depend only on the constantc B of the inequality in Lemma 4.2. The proof is completely analogous to the proof for the continuous case and is, therefore, omitted. The only difference is the use of the estimate from Lemma 4.2 instead of Friedrichs inequality. So, in short, with V h = π h (S h,0 ) H h (divdiv h,) H h (divdiv h,) = {τ V h : divdiv h τ,v h = 0 for allv h Q h }. For describing the space H h (divdiv h,) more explicitly, we consider the subspace of all functions in H (divdiv,) which can be represented by a finite element function φ (S h ) 2, for which we show the following result. THEOREM 5.2. Letbe simply connected. Then H h (divdiv h,) = {H T ε(φ)h: φ (S h ) 2 }.

16 14 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER Proof. Let φ (S h ) 2. Then τ = H T ε(φ)h P k 1 for all triangles T T h. Furthermore, let e be an edge of a triangle T with outer unit normal vector n = (n 1,n 2 ) T and unit tangent vectors = ( n 2,n 1 ) T. By elementary computations we obtain τ nn = n T H T ε(φ)hn = s s φ. So, τ nn depends only on values of φ on the edge e, which immediately implies that τ nn is continuous on inter-element boundaries. This shows that τ lies in V h, and, therefore, the inclusion{τ = H T ε(φ)h: φ (S h ) 2 } H h (divdiv h,) follows. The equality follows by comparing the dimensions. We have dim{τ = H T ε(φ)h: φ (S h ) 2 } = 2dimS h dim RM = 2dimS h 3. On the other hand, by Theorem 5.1, it follows that dim H h (divdiv h,) = dimv h dims h,0. A simple count of the degrees of freedom forv h yields dimv h = dims h,0 +2 dims h 3. Therefore, H h (divdiv h,) = 2 dims h 3, which completes the proof. REMARK 5.3. A consequence of the last theorem is the important inclusion H h (divdiv h,) H (divdiv,), which resembles the corresponding result of Lemma 5 in [17]. Therefore, we have the following representation of the approximate solutionσ h V h of (4.2): (5.1) σ h = π h (p h )+H T ε(φ h )H. The analogous representation for the test functions τ = π h (q) + H T ε(ψ)h leads to the following equivalent formulation of (4.2). Find p h S h,0, φ h (S h ) 2 /RM, w h S h,0 such that (5.2) ˆπ h (p h ) : ˆπ h (q)dx+ ˆπ h (p h ) : ε(ψ) dx + p h v dx for allq S h,0,ψ (S h ) 2 /RM,v S h,0, and with ˆπ h (q) : ε(φ h ) dx+ w h q dx = 0, ε(φ h ) : ε(ψ) dx = 0 ˆπ h (q) = Hπ h (q)h T. = f,v Observe that the HHJ method in the form of (5.2) is a non-conforming method for (3.3), while (4.1) can be seen as a conforming variant. Compared to (5.2), the conforming variant is slightly less costly, since the linear operatorπ h is not needed.

17 THE HELLAN-HERRMANN-JOHNSON METHOD Error estimates. Since the new mixed variational formulation is equivalent to the original primal variational formulation (without any additional assumption on ) and the finite element space V h of the original Hellan-Herrmann-Johnson method has not changed (but only its representation, which amounts to a change of basis), all known error estimates for the original Hellan-Herrmann-Johnson method are still valid. For completeness we briefly recall some of the most important estimates from [17], [21], [3], and [9], and give a sketch of the proofs using the framework of the new variational formulation. Afterwards we will present new error estimates for the conforming variant (4.1) The original Hellan-Herrmann-Johnson method. We closely follow the approach in [3], which is based on two essential assumptions, regularity and consistency. Regularity is ensured in [3] by considering only convex domains. Consistency is ensured in [3] by setting up a framework in which the Hellan-Herrmann-Johnson method becomes a conforming method. Then the estimates easily follow from the existence of two interpolation operators Π h andi h forσ andw, respectively, and their approximation properties. Here we recall a (weaker) regularity result from [8, 9], which is valid without any additional assumption on. THEOREM 6.1. If f H 1 (), then the solution w of (2.1) lies in W 3,p () for some p (4/3,2], and there is a constantc > 0 such that (6.1) w W 3,p () c f 1. Actually, the regularity results in [8, 9] cover a much larger class of boundary conditions. Although the Hellan-Herrman-Johnson method is a non-conforming method with respect to (2.10), we still have consistency. In order to show this, we need to extend the bilinear form b h, originally defined onv h Q h by b h (τ,v) = { } τ : 2 v dx τ nn n v ds, T T T T h to the larger domainṽ Q, given by and Ṽ = {τ L 2 () sym : τ ij T W 1,p (T) for all T T h, 1 i,j 2, and τ nn is continuous across inter-element boundaries} Q = {v H 1 0 (): v T H 2 (T) for all T T h }. The well-definedness ofb h onṽ Q follows from standard embedding theorems for Sobolev spaces; see, e.g., [1]. Observe that Ṽ coincides with the space in (4.3) forp = 2. Therefore, we keep the same notation. Now we can show the following consistency result. THEOREM 6.2. For the solution(σ,w) of (2.10), we have (6.2) a(σ,τ) +b h (τ,w) = 0 for all τ Ṽ, b h (σ,v) = f,v for all v Q. Proof. From Theorem 6.1 it follows that b h (τ,w) and b h (σ,v) are well-defined for all (τ,v) Ṽ Q.

18 16 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER We have b h (τ,w) = T T h = T T h { T T τ : 2 w dx T τ : 2 w dx = } τ nn n w ds τ : 2 w dx, sinceτ nn is continuous and n w changes sign across interelement boundaries, and n w = 0 on. With 2 w = σ, the first line in (6.2) follows. Moreover, we have b h (σ,v) = T T h T T h { T } σ : 2 v dx σ nn n v ds T = { (divσ) v dx T T T h = { (divσ) v dx T T T } (σn) v ds+ σ nn n v ds T } σ ns s v ds = T T h T (divσ) v dx, since s v is continuous and σ ns changes sign across interelement boundaries, and v = 0 on. Therefore, b h (σ,v) = (divσ) v dx = divdivσ,v, where the last identity follows from the corresponding identity divdivσ,v = σ : 2 v dx = (divσ) v dx for all v C0 (), by a continuity argument in H0 1 (). With divdivσ = f the second line in (6.2) follows, which completes the proof. Having again regularity and consistency we proceed as in [17], [3]. The interpolation operator Π h is defined on Ṽ by the conditions (4.4) and (4.5) for τ h = Π h τ V h and τ Ṽ. The second interpolation operatori h is given by the conditions v h (a) = v(a) for all verticesaoft, T T h, v h q ds = vq ds, for all q P k 2, for all edgese oft, T T h, e e v h q dx = vq dx, for all q P k 3, T T h, T T for v h = I h v Q h and v Q. It is easy to see that the two interpolation operators satisfy the following properties. b h (τ Π h τ,v h ) = 0 for all τ Ṽ, v h Q h, b h (τ h,v I h v) = 0 for all τ h V h, v Q. The rest of the arguments are identical to the arguments used in [3] with a straightforward adaptation to the weaker regularity condition (6.1), where it is needed. This leads directly to the following known estimate forσ, see [3]. THEOREM 6.3. Let(σ,w) and(σ h,w h ) solve (2.10) and (4.2), respectively. Then σ h σ 0 σ Π h σ 0.

19 THE HELLAN-HERRMANN-JOHNSON METHOD 17 The error estimates for w read as follows. For completeness we give a sketch of the proof, which closely follows the arguments in [3]. THEOREM 6.4. Assume that (6.1) is satisfied for some p (4/3,2]. Then there is a constantc > 0 such that { ch w 3,p fork = 1, w h w 1 w I h w 1 + ch 2 2/p σ Π h σ 0,h fork 2, with the mesh-dependent norm τ 0,h, given by τ 2 0,h = τ 2 0 +h e E h (τ) nn 2 L 2 (e). Here E h denotes the set of all edges of triangles from T h. Proof. By the triangle inequality we have w h w 1 w I h w 1 + I h w w h 1. For estimating I h w w h 1 a duality trick is used. Ford H 1 (), let w d H0() 2 be the unique solution to (6.3) 2 w d : 2 v dx = d,v for allv H0 2 (), and setσ d = 2 w d. Then it follows from Theorem 6.1 thatw d W 3,p (),σ d W 1,p (), and a(σ d,τ)+b(τ,w d ) = 0 for allτ H 1 (divdiv,) sym, b(σ d,v) = d,v for allv H 1 0 (). From the consistency result in Theorem 6.2 applied to this problem it follows that: a(σ d, τ) +b h ( τ,w d ) = 0 for all τ Ṽ, b h (σ d,ṽ) = d,ṽ for all ṽ Q. In particular, for τ = σ σ h andṽ = I h w w h we obtain a(σ d,σ σ h ) +b h (σ σ h,w d ) = 0, b h (σ d,i h w w h ) = d,i h w w h. Using Galerkin orthogonality it follows that d,i h w w h = a(σ d,σ σ h )+b h (σ σ h,w d )+b h (σ d,i h w w h ) = a(σ d,σ σ h )+b h (σ σ h,w d )+b h (σ d,w w h )+b h (σ d,i h w w) = a(σ d Π h σ d,σ σ h )+b h (σ σ h,w d I h w d )+b h (σ d Π h σ d,w w h ) +b h (σ d,i h w w). The properties of the interpolation operators imply b h (σ σ h,w d I h w d ) = b h (σ Π h σ,w d I h w d )

20 18 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER and Therefore, which implies b h (σ d Π h σ d,w w h ) = b h (σ d Π h σ d,w I h w) b h (σ d,i h w w) = b h (σ d Π h σ d,i h w w). d,i h w w h = a(σ d Π h σ d,σ σ h )+b h (σ Π h σ,w d I h w d ), d,w h I h w σ d Π h σ d 0 σ σ h 0 + b h (σ Π h σ,w d I h w d ) using Theorem 6.3. We have σ d Π h σ d 0 σ Π h σ 0 + b h (σ Π h σ,w d I h w d ) b h ( τ,ṽ) τ 0,h ṽ 2,h with a second mesh-dependent norm v 2,h, given by v 2 2,h = T T h v 2 H 2 (T) + 1 h e E h [ n v] e 2 L 2 (e), where[ n v] e denotes the jump of n v across an edge, see [3, (4.48)]. Then it follows that w h I h w 1 = sup d H 1 () [ sup d H 1 () d,w h I h w d 1 σ d Π h σ d 0 + w d I h w d 2,h d 1 ch 2(1 1/p) σ Π h σ 0,h. using the interpolation error estimates and the regularity estimates σ d Π h σ d 0 ch 2 2/p σ d 1,p, w d I h w d 2,h ch 2 2/p w d 3,p fork 2, σ d 1,p c d 1 and w d 3,p c d 1. ] σ Π h σ 0,h Fork = 1 we have b h (σ Π h σ,w d I h w d ) = b h (σ,w d I h w d ) = (divσ) (w d I h w d ) dx. Therefore, b h (σ Π h σ,w d I h w d ) divσ 0,p w d I h w d 1,q c σ 1,p w d I h w d 1,q c w 3,p w d I h w d 1,q withq = p/(p 1). Now w d I h w d 1,q ch w d 2,q ch w d 3,p ch d 1,

21 THE HELLAN-HERRMANN-JOHNSON METHOD 19 which implies d,w h I h w w h I h w 1 = sup ch w 3,p. d H 1 () d 1 This completes the proof. Using standard approximation properties of the interpolation operators Π h and I h (see [17], [3], and [9]) one immediately obtains the following consequences. COROLLARY 6.5. If w H k+2 (), then we have estimates of optimal order forσ. σ σ h 0 ch k σ k. COROLLARY 6.6. Assume that (6.1) is satisfied for some p (4/3,2]. 1. Forf H 1 () we have σ σ h 0 ch 2 2/p f 1 and w w h 1 ch min(k,4 4/p) f Ifw H k+1 (), then we have w w h 1 ch k+1 2/p w k+1 fork Ifis convex andw H k+1 (), then we have estimates of optimal order forw. w w h 1 ch k w k+1 fork 2. The first estimates in Corollary 6.6 are in accordance with [9]. Observe that 2 2/p > 1/2 and 4 4/p > 1, since p > 4/3. Therefore, we have convergence rates at least of order O(h 1/2 ) ando(h) for σ σ h 0 and w w h 1, respectively. Observe thatk+1 2/p > k 1/2, since p > 4/3. Therefore, we have a convergence rate at least of order O(h k 1/2 ) for w w h 1, ifw H k+1 (). Ifis convex, then (6.1) holds forp = 2, see [9] The conforming variant (4.1). The only difference to the original HHJ method is the use of the finite element space V conf h = {τ = π(q)+h T ε(ψ)h: q S h,0, ψ (S h ) 2 /RM} instead of V h for approximating the Lagrangian multiplierσ. As before the error analysis is based on two interpolation operatorsπ conf h andih conf satisfying withi conf h b(τ Π conf h τ,v h ) = 0 for allτ H 1 (divdiv,) sym, v h Q h, b(τ h,v Ih conf v) = 0 for allτ h Vh conf, v H0(), 1 = R h, andπ conf h is given by Π conf h τ = π(r hq)+h T ε(ĩhφ)h for τ = π(q)+h T ε(φ)h. HereR h : H0 1() S h,0 denotes the Ritz projection, given by R h v q h dx = v q h dx for all v H0 1 (), q h S h,0, and Ĩh: H 1 () 2 /RM S 2 h /RM can be any reasonable interpolation operator like a Clément-type interpolation operator, see, e.g., [22]. It turns out that the Ritz projection is the

22 20 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER only candidate forih conf. So, note that we have again the same properties of these interpolation operators. However, these operators are no longer local operators as they were in the case of the original HHJ methods. That does not effect the analog of Theorem 6.3 but it leads to a deterioration of the estimates for the analog of Theorem 6.4 and the subsequent corollaries for non-convex domains. For convex domains the interpolation operators share the same approximation properties as before resulting in the following error estimates. THEOREM 6.7. Let be convex. 1. Ifw H k+2 (), then we have estimates of optimal order forσ. σ σ h 0 ch k σ k. 2. Ifw H k+1 (), then we have estimates of optimal order forw. w w h 1 ch k w k+1 fork 2. The proof of these estimates is a complete copy of the corresponding proofs for the original HHJ method, however this time based on the interpolation operators introduced above, and is, therefore, omitted. Using the same techniques for estimating the errors in the nonconvex case, leads to a deterioration of the estimates for two reasons. First of all, standard L 2 -error estimates for the Ritz projection are not of optimal approximation order, and, secondly, smoothness assumptions on σ do not completely result in the corresponding smoothness assumption forp(andφ) due to the lack of full elliptic regularity of (3.1). 7. Numerical experiments. The obvious procedure for solving (5.2) consists of three consecutive steps. step 1. For givenf H 1 (), solve (7.1) p h v dx = f,v by the preconditioned conjugate gradient (PCG) method with a standard multigrid preconditioner for a Poisson problem. step 2. Forp h, computed in step 1, solve (7.2) ε(φ h ) : ε(ψ) dx = ˆπ h (p h ) : ε(ψ) dx by the PCG method with a standard multigrid preconditioner for a pure traction problem. Observe that the bilinear form in (7.2) has a non-trivial kernel, namely RM. See, e.g., [24] for the treatment of such singular problems. We have chosen the alternative approach of regularizing the problem by replacing the original bilinear form by ε(φ h ) : ε(ψ) dx+ φ h dx ψ dx+ curlφ h dx curlψ dx, which results in a coercive problem in Sh 2 with respect to theh1 -norm. step 3. Forp h andφ h, computed in step 1 and 2, respectively, solve (7.3) w h q dx = ˆπ h (p h ) : ˆπ h (q) dx ˆπ h (q) : ε(φ h ) dx by the PCG method with a standard multigrid preconditioner for a Poisson problem.

23 THE HELLAN-HERRMANN-JOHNSON METHOD 21 For the conforming variant (4.1), the right-hand sides in (7.2) and (7.3) have to be replaced by the simpler expressions π(p h ) : ε(ψ) dx = p h divψ dx and π(p h ) : π(q)dx π(q) : ε(φ h ) dx = 2 p h q dx q divφ h dx, respectively. For each of the three multigrid preconditioners we choose one multigrid V-cycle with one forward and one backward Gauss-Seidel sweep for pre- and post-smoothing, respectively. In each of the three steps the initial guess for the PCG method is set to 0. We will now discuss the accuracy and computational complexity of this procedure in more detail. It is well-known that the multigrid V-cycle algorithm described above converges with a convergence rate in energy norm which is bounded below 1 uniformly with respect to the mesh size h, see [11]. Therefore, the number of PCG iterations that are necessary to reduce an initial error by a prescribed factor, say δ > 0, in energy norm, is uniformly bounded. In particular, the number of PCG iterations necessary to obtain an approximate solution p h to (7.1), such that (7.4) p h p h 1 δ p h 1, is uniformly bounded. In the second step the PCG method is applied to (7.2), however with p h (which is not available) replaced by p h on the right-hand side. Let φ h denote the exact solution of this modified problem. Then the number of PCG iterations necessary to obtain an approximate solution φ h to the modified problem in step 2, such that (7.5) ε( φ h φ h ) 0 δ ε( φ h ) 0, is uniformly bounded, too. Analogously, in step 3 the number of PCG iterations necessary to obtain an approximate solution w h to (7.3), however with p h and φ h replaced by p h and φ h on the right-hand side, respectively, such that (7.6) w h w h 1 δ w h 1, is uniformly bounded, where w h denotes the exact solution to the modified problem in step 3. From p h and φ h we obtain an approximation σ h given by σ h = π h ( p h )+H T ε( φ h )H. Now we have LEMMA 7.1. Assume that (7.4), (7.5), and (7.6) hold for someδ δ 0. Then (7.7) σ h σ h 0 c 1 δ f 1 and w h w h 1 c 2 δ f 1 with positive constantsc 1 andc 2 which depend only on δ 0 and the constantc B from Lemma 4.2. Proof. We first estimate the difference between the solutions of the original and the modified problems in step 2 and 3 in terms of the difference of the data on the right-hand sides.

24 22 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER Subtracting (7.2) from its modified variant forψ = φ h φ h yields ε( φ h φ h ) 2 0 = ˆπ h ( p h p h ) : ε( φ h φ h ) dx c B p h p h 1 ε( φ h φ h ) 0, where Lemma 4.2 was applied. This implies (7.8) ε( φ h φ h ) 0 c B p h p h 1. Analogously, subtracting (7.3) from its modified variant forq = w h w h yields w h w h 2 1 = ( σ h σ h ) : π h ( w h w h ) dx c B σ h σ h 0 w h w h 1, which implies (7.9) w h w h 1 c B σ h σ h 0. Then we have σ h σ h 0 π h ( p h p h ) 0 + ε( φ h φ h ) 0 c B p h p h 1 + ε( φ h φ h ) 0 + ε( φ h φ h ) 0 2c B p h p h 1 + ε( φ h φ h ) 0 2c B δ p h 1 +δ ε( φ h ) 0, using (7.4), and (7.5) for the last estimate. Next we estimate ε( φ h ) 0. We have using (7.8) and (7.4) and ε( φ h ) 0 ε( φ h φ h ) 0 + ε(φ h ) 0 c B δ p h 1 + ε(φ h ) 0 ε(φ h ) 0 = H T ε(φ h )H 0 = σ h π h (p h ) 0 c B p h 1 + σ h 0 using (5.1) and Lemma 4.2, which imply ε( φ h ) 0 c B (1+δ) p h 1 + σ h 0. Using this estimate we obtain σ h σ h 0 c B δ(3+δ) p h 1 +δ σ h 0 δ ( 1+[(3+δ)c B ] 2) 1/2 ( ph σ h 2 0 = δ ( 1+[(3+δ)c B ] 2) 1/2 σh 1,divdiv,h. Furthermore, using (7.6) we obtain ) 1/2 and, therefore, w h w h 1 w h w h 1 + w h w h 1 δ w h 1 + w h w h 1 δ w h 1 +(1+δ) w h w h 1. w h w h 1 δ w h 1 +c B (1+δ) σ h σ h 0

25 THE HELLAN-HERRMANN-JOHNSON METHOD 23 using (7.9). From the stability estimates for saddle point problems (see, e.g., [31, Theorem 1]) σ h 1,divdiv,h 1 β a α f 1 and w h 1 a β 2 a α f 1 with a = α = 1 andβ = (1+c 2 B ) 1/2 (see Theorem 4.4), we finally obtain (7.7) with c 1 = ( 1+[(3+δ 0 )c B ] 2) 1/2( 1+c 2 B ) 1/2, c2 = 1+c 2 B +c B (1+δ 0 )c 1. This lemma eventually shows that the number of PCG iterations necessary to obtain approximate solutions for w h and σ h with a tolerance of order δ relative to the data f 1 of the problem is bounded independently of the mesh size h. With respect to this criterion of accuracy the proposed procedure is of optimal computational complexity, since the number of arithmetic operations for applying one multigrid V-cycle is proportional to the number of involved unknowns and the number of PCG iterations is uniformly bounded. As expected no additional smoothness of the data f is required for these arguments. In case that δ is not considered as a prescribed fixed quantity but is chosen proportional to the order of the discretization error, full multigrid methods have to be considered instead to restore optimal computational complexity. REMARK 7.2. An estimate of the form (7.7) can also be shown for the conforming variant with a completely analogous proof. In order to illustrate the theoretical results we consider the following simple biharmonic test problem: 2 w = f in, w = n w = 0 onγ on two domains, the square = S = ( 1,1) 2 and thel-shaped domain = L depicted in figures 7.1 and 7.2, where also the initial meshes (level l = 0) are shown. (The initial meshes were created by distorting a originally uniform subdivision of S into 32 and L into 24 triangles, in order to avoid any artificial super-convergence effects due to uniformity.) The right-hand side f(x) is chosen such that w(x) = [ 1 cos(2πx 1 ) ][ 1 cos(4πx 2 ) ] is the exact solution to the problem. The initial meshes are uniformly refined until the final level l = L. In all experiments the polynomial degree k as introduced in the beginning of Section 4 is chosen equal to 1, which represents the lowest order HHJ method. FIG = S. FIG = L. In each of the three steps, a reduction of the Euclidean norm of the initial residual by a factor of10 8 was used as stopping criterion for the PCG methods. Table 7.1 shows the observed number of iterations for the solution procedure as described above for = S. The first column contains the levellof refinement. The next three pairs

26 24 W. KRENDL, K. RAFETSEDER, AND W. ZULEHNER of columns show the total numbern i of degrees of freedom and the number of iterations iter i of the PCG method for the linear system in step i = 1,2,3. TABLE 7.1 Number of iterations, = S (square). L N 1 iter 1 N 2 iter 2 N 3 iter Table 7.2 shows the corresponding results for thel-shaped domain = L representing a non-convex case. TABLE 7.2 Number of iterations, = L (L-shaped domain). L N 1 iter 1 N 2 iter 2 N 3 iter As expected the number of PCG iterations is bounded uniformly with respect to the mesh size. Finally, in Tables 7.3 and 7.4 the discretization errors of the original HHJ method (5.2) and its conforming variant (4.1) are shown. For the original HHJ method, theh 1 -error of the original variable w and the L 2 -error of the auxiliary variableσ decrease with the order h, in accordance with known estimates, see Section 6. The last two columns contain the errors of p and φ, given by (3.2) and measured in the associated norms of the Helmholtz-decomposition, see Theorem 3.1 and 3.3, where the (analytically not available) exact solutions p and φ are replaced by their approximate solutions on level l = 10. The conforming variant behaves similarly, see Table 7.4. TABLE 7.3 Discretization errors for the HHJ method (5.2), = L (L-shaped domain). L w w h 1 σ σ h 0 p p h 1 φ φ h