A POSTERIORI ERROR ESTIMATES FOR FOURTH-ORDER ELLIPTIC PROBLEMS

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1 A POSTERIORI ERROR ESTIMATES FOR FOURTH-ORDER ELLIPTIC PROBLEMS Slimane Adjerid Department of Mathematics and Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg, VA Abstract We extend the dichotomy principle of Babuska and Yu [26, 27] and Adjerid et al. [3, 4] for estimating the finite element discretization error to fourth-order elliptic problems. We show how to construct a posteriori error estimates from jumps of the third partial derivatives of the finite element solution when the finite element space consists of piecewise polynomials of odd degree and from the interior residuals for even degree approximations on meshes of square elements. These estimates are shown to converge to the true error under mesh refinement. We also show that these a posteriori error estimates are asymptotically correct for more general finite element spaces. Computational results from several examples show that the error estimates are accurate and efficient on rectangular meshes. 1 Introduction A posteriori error estimators and/or indicators are an integral part of adaptive finite element methods for partial differential equations. They are used for secondorder elliptic [11, 9, 10, 25, 26, 27] and parabolic [3, 7, 4, 8] problems. More recently a posteriori error estimates have been developed for hyperbolic problems [23, 5, 6]. Local error estimates are used to control adaptive enrichment by refining regions having large errors and coarsening regions with small errors. On the other-hand, global a posteriori error estimates are used to control solution accuracy. Ideal a posteriori error estimates should be computationally simple and provide guaranteed asymptotical exactness in the sense that the estimates should converge to the true errors from above as the finite element space is enriched. Even-odd error estimates have been used successfully by Yu [26, 27] for elliptic problems and Adjerid et. al [3, 4] for parabolic problems. They are asymptotically correct for tensor product [26, 27, 3] and hierarchical [4] finite element spaces under mesh refinement. 1

2 Most of the existing error estimators are applicable to second-order problems. However, many important physical situations, such as elasticity [22], fluid mechanics [17] and thin film growth [20, 28] are modeled by boundary-value problems of fourth-order. The first a posteriori error estimate for fourth-order problems appeared in Verfurth [25]. The author is not aware of any other attempt to compute efficient a posteriori error estimates for higher-order problems. In this paper we show how to extend the even-odd error estimation procedures to linear fourth-order boundary value problems. We use C 1 finite element approximations on a square mesh and compute error estimates using jumps of the third partial derivatives of odd-degree finite element solutions and by solving local residual problems for even-degree solutions. The even-odd error estimates are both very simple and efficient to compute. Furthermore, the even-degree error estimators do not involve any neighbor information, which makes them ideal for parallel computations. Moreover, Numerical results suggest that even-degree estimators perform better than odd-degree estimators. Both even- and odd-degree estimators are extended to more general hierarchical finite element spaces. More recent approaches to a posteriori error estimation include guaranteed estimates [12, 13], weighted and goal-oriented estimates [14, 19] and bound estimates for functionals [21]. Here we will show that our estimates are asymptotically correct in the H 2 norm. In section 3, we prove several lemmas. We establish asymptotic exactness of these error estimates on square elements for both bi p even- 4 and odd-degree 5. These error estimates are further extended to the more economical hierarchical approximations in 6. We test our error estimators on two examples in 7 and present numerical results which show that the actual theory holds for square- and rectangular-element meshes. Finally, numerical results [4] suggest that the even-odd error estimators are applicable to more general meshes as well as problems with singularities and we expect this to continue to hold for fourth-order problems. 2 Problem formulation Let us consider the linear scalar fourth-order partial differential equation Lu = f, x = (x 1,x 2 ) Ω = [0, 1] 2, (2.1a) with Lu = ( 1) β D β (a α,β (x)dα u), (2.1b) α 2 β 2 where α = α 1 + α 2, β = β 1 + β 2 and D α = α 1 x 1 α 2 x 2, (2.1c) subject to the Dirichlet boundary conditions u(x) = 0, u (x) = 0, x Ω, (2.1d) ν 2

3 where u ν is the outward normal derivative of u. The functions a α,β (x), α, β 2, are smooth functions and L is a positive definite operator on the Sobolev space H0 2. The Galerkin form of (2.1) consists of determining u H0 2 satisfying A(v, u) = (v, f), v H 2 0, (2.2a) where the strain energy and L 2 inner products, respectively, are A(v, u) = a α,β Dα ud β v dx 1 dx 2 Ω α 2 β 2 (2.2b) and (v, u) = Ω uvdx 1 dx 2. (2.2c) Sobolev spaces H s, s 0 are equipped with the usual inner product, norm and seminorm (v, u) s = (D α v, D α u), (2.3a) α s u 2 s = (u, u) s, (2.3b) u 2 s = (D α u, D α u), (2.3c) α =s wherea0subscriptonh 2 implies that functions also satisfy (2.1d). Finite element solutions of (2.2a) are obtained by approximating H0 2 dimensional subspace S N,p 0 and determining U S N,p 0 such that by a finite A(V,U) = (V,f), V S N,p 0. (2.4) Let us partition Ω into a uniform mesh of square elements i,i= 1, 2,,N and define S N,p as S N,p = {w H 2 w(x) Q p ( i ), x i, i = 1, 2,..., N }, (2.5) where Q p ( i ) is the space of bi-polynomial functions that are products of univariate polynomials of degree p in x 1 and x 2 on i. Again with the subscript 0 S N,p 0 = S N,p H0 2. In the following lemmas we describe standard interpolation and aprioridiscretization error estimates for conforming finite element solutions of (2.1) that will be used in the subsequent analysis. 3

4 Lemma 1. Let W S N,p 0 be an interpolant of w H 2 0 Hp+1 that is exact when w Q p (Ω). Then, there exists a constant C > 0 such that where W w s Ch p+1 s w p+1, s = 0, 1, 2, (2.6a) h = 1/ N. (2.6b) Further, let u H 2 0 Hp+1 and U S N,p be solutions of (2.1) and (2.4), respectively. Then, there exists a constant C > 0 such that u U s Ch p+1 s u p+1, s = 0, 1, 2. (2.6c) Proof. Cf. Oden and Carey [18]. We also recall the trace theorem Lemma 2. Let Ω be a bounded open set of R n with a Lipschitz boundary Ω. Then there exists a constant, C > 0, such that v 0, Ω C(h 1/2 v 0,Ω + h 1/2 v 1,Ω ), v H 1. (2.7) Proof. Cf. Grisvard [16]. Lemma 3. Let W P p () and its mapping to the canonical element Ŵ P p ([ 1, 1] 2 ). Then there exist constants C 1, C 2 > 0 such that and C 1 Ŵ s+1 < Ŵ s < C 2 Ŵ s+1, Ŵ P p([ 1, 1] 2 ), s 0 (2.8) C 1 W s+1 < h W s < C 2 W s+1, W P p (), s 0, (2.9) where P p () is the space of polynomials of degree p on an element Proof. (2.8) is obtained from the equivalence of all norms on finite dimensional spaces. Similarly, (2.9) is straight forward using the mapping of the square element [ z h/2, z + h/2] 2 to the canonical element [ 1, 1] 2. Let us construct a hierarchical basis for S N,p on the canonical element [ 1, 1] 2 as a tensor product of the one dimensional hierarchical basis consisting of the four standard cubic Hermite polynomial shape functions Nl m, l = 1, 1, m = 0, 1, complemented by the interior shape functions N0 i, i 4, such that N 1 0 ( 1) = 1, N0 1 (1) = 0, dn 1 0 (±1) = 0, dx (2.10a) N 1 1(±1) = 0, dn 1 1 dn ( 1) = 1, dx 1 1 (1) = 0, dx (2.10b) 4

5 N 0 1 ( 1) = 0, N 0 1 (1) = 1, dn 0 1 dx (±1) = 0, (2.10c) N1 1 dn1 1 (±1) = 0, dx ( 1) = 0, dn1 1 (1) = 1. dx (2.10d) Higher-order interior shape functions are obtained by integrating Lobatto polynomials on [ 1, 1] to obtain N i 0(ξ) = ξ 1 P i 1 (x)dx, i = 4, 5,..., p, (2.11) where P i is the Lobatto polynomial of degree i P i (x) = P i(x) P i 2 (x) 2(2i 1), (2.12) with P i being the Legendre polynomial of degree i. Using the orthogonality of Legendre polynomials yields ( N0 i+1 1 Pi+1 = 2i 1 2i +1 P ) i 1 2i 3 = (2i 3)P i+1 2(2i 1)P i 1 +(2i +1)P i 3. (2.13) 2(2i 1)(4i 2 4i 3) One can easily verify that the shape functions N0 i, i 4, satisfy (i) 1 d 2 N0 i 1 d 2 N j dξ 2 0 dξ 2 dξ = δ ij, i,j 4, where δ ij is the Kronecker symbol. (ii) N i 0 has i 4 simple roots in ( 1, 1) and two double roots at ±1. (iii) N0 i = (ξ 2 1) 2 Ñ i 4 (ξ), where Ñ 2l has 2l non-zero roots in ( 1, 1) symmetrically disposed with respect to the origin, {±ξ k, k =1, 2,, l}. On the other-hand Ñ 2l+1 has 2l + 1 roots in ( 1, 1), {0} {±ξ k,k =1, 2,, l}. The two-dimensional basis on [ 1, 1] [ 1, 1] is defined by 16 vertex modes: N k,m ij (ξ,η) = N k i (ξ)n m j (η), i,j = 1, 1, k, m =0, 1. (2.14a) 16(p-3) Edges modes: N j,n i,1 = N j i (ξ)n 0 n (η), i = 1, 1, j = 0, 1, n = 4,..., p, (2.14b) 5

6 N j,n i,2 = N j i (η)n n 0 (ξ), i = 1, 1, j = 0, 1, n = 4,..., p, (2.14c) N j,n i,3 = N j i (ξ)n 0 n (η), i = 1, 1, j = 0, 1, n = 4,..., p, (2.14d) N j,n i,4 = N j i (ξ)n n 0 (η), i = 1, 1, j = 0, 1, n = 4,..., p, (2.14e) (p 3) 2 Interior modes: N j,k 0 = N j 0 (ξ)n 0 k (η), j,k = 4,...,p. (2.14f) 3 Preliminary considerations Let π[ z] be the univariate operator that interpolates functions in H0 2 ( z h/2, z+ h/2) and their first derivatives such that πu( z ± h/2) = u( z ± h/2), (3.1a) d πu dx du ( z ± h/2) = ( z ± h/2), dx (3.1b) πu( z ± ξ j h/2) = u( z ± ξ j h/2), j = 1, 2,... s, for p 3=2s (3.1c) or πu( z ± ξ j h/2) = u( z ± ξ j h/2), j = 0, 1, 2,... s, for (p 3) = 2s +1, (3.1d) where ξ j are the simple roots of N p+1 0. Thus, for p 3, the function ψ p+1 ( z,z) = z p+1 π[ z]z p+1 (3.2) and N p+1 0 have identical roots with first derivatives which both vanish at z ± h/2. Hence, ψ p+1 ( z,z), ψ p+1 ( z,z), and ψ p+1 ( z,z) are, respectively, proportional to N p+1 0, Lobatto and Legendre polynomials on [ z h/2, z + h/2], with ( ) denoting ordinary differentiation. We use π to define a two-dimensional interpolation operator π i on element i satisfying π i u(x) = π[ x 1,i ] π[ x 2,i ]u(x) Q p ( i ), x i. Since the mesh is uniform, we will omit the elemental index i and the dependence of π[ x j,i ]and ψ( x j,i,x i ) on the coordinates of the cell center ( x 1,i, x 2,i ), i = 1, 2,..., N, whenever confusion is unlikely. 6

7 The functions ψ p+1 (x j ), j = 1, 2, provide a basis for the dominant contribution to the spatial discretization error on element for both odd- and even-order finite element approximations. Indeed, we shall show that estimates E(x) ofe(x) have the form E(x) = b 1 ψ p+1 (x 1 ) + b 2 ψ p+1 (x 2 ), x. In order to estimate the finite element discretization error, we need to prove the following series of lemmas. Lemma 4. Let πu be an interpolant of u H p+2,then where u(x) πu(x) = φ(x) + γ(x), x, (3.3a) φ(x) = β 1 ψ p+1 (x 1 ) + β 2 ψ p+1 (x 2 ), (3.3b) D α φ 0, Ch p+1 α u p+1,, α 2, (3.3c) γ s, Ch p+2 s u p+2,, s = 0, 1,..., p + 1, (3.3d) and xj γ s, Ch p+1 s u p+2,, s = 0, 1, j = 1, 2, (3.3e) 2 x j ψ p+1 (x j )x k j dx 1dx 2 = 0, k = 0, 1,..., p 2, p 3. (3.3f) Remark. Local Sobolev norms are defined like their global counterparts (2.3b) and (2.3c) with Ω replaced by. Proof. Let πu Q p and V = w + β 1 x p β 2 x p+1 2 such that w Q p, πv = πu and V interpolates u at two other points. Then the interpolation error can be written as Using the fact that πu = πv we have Thus u πu = u V + V πu. (3.4) u πu = u V + V πv = u V + β 1 (x p+1 1 π[ x 1 ]x p+1 1 ) + β 2 (x p+1 2 π[ x 2 ]x p+1 2 ). (3.5) u πu = γ + φ, (3.6) where γ = u V. Using standard interpolation error estimates (2.6a) we obtain (3.3c-3.3e). Finally, we note that ψ p+1 and N p+1 0 are polynomials of degree p +1 with identical roots and therefore are proportional, ψ p+1 N p+1 0, p 3. This leads to the orthogonality result (3.3f). 7

8 Lemma 5. Let Πu S N,p 0 be an interpolant of u H p+2 for x Ω that agrees with πu when x, then A(W, u Πu) Ch p u p+2 W 2, W S N,p 0. (3.7) Proof. Let us consider first a (2,0),(2,0) (x) x 2 1 W x 2 1 (u πu)dx 1 dx 2 = ā (2,0),(2,0) 2 x 1 W 2 x 1 (u πu)dx 1 dx 2 + ( a(2,0),(2,0) (x) ā (2,0),(2,0) ) 2 x1 W 2 x 1 (u πu)dx 1 dx 2, (3.8) where ā (2,0),(2,0) = a (2,0),(2,0) ( x), with x being the center of the element. Using the continuity of a α,β and applying the Schwarz inequality with the estimates (2.6a) we have ( ) a(2,0),(2,0) (x) ā (2,0),(2,0) 2 x1 W x 2 1 (u πu)dx 1 dx 2 < Ch u Πu 2, W 2, < Ch p u p+2, W 2,. (3.9) Using (3.3a) and (3.3b), the first term in the right-hand side of (3.8) becomes x 2 1 W x 2 1 (u πu)dx 1 dx 2 = x 2 1 W ( x 2 1 φ + x 2 1 γ)dx 1 dx 2. (3.10) Using (3.3b) and the fact that ψ p+1 is proportional to the Legendre polynomial P p 1 we have x 2 1 W x 2 1 φdx 1 dx 2 = 0, W S N,p. (3.11) Applying the Schwarz inequality and using (3.3e) leads to x 2 1 W x 2 1 γdx 1 dx 2 Ch p u p+2 W 2, W S N,p. (3.12) Following the same steps we prove ( 2 x 2 W, 2 x 2 (u πu)) Ch p u p+2, W 2, W S N,p. (3.13) Applying Green s theorem we obtain x 2 2 W x 2 1 (u πu)dx 1 dx 2 = Γ 1 x2 W 2 x 1 (u πu)dx 1 8 Γ + 1 x2 W 2 x 1 (u πu)dx 1

9 Using (3.3a) we have x2 W 2 x 1 x2 (u πu)dx 1 dx 2. (3.14) x2 W 2 x 1 x2 (u πu)dx 1 dx 2 = Using the fact that x 2 1 x2 φ = 0 we have x2 W x 2 1 x2 (u πu)dx 1 dx 2 = x2 W 2 x 1 x2 (φ + γ)dx 1 dx 2. (3.15) x2 W 2 x 1 x2 γdx 1 dx 2. (3.16) Integration by parts yields x 2 W x 2 1 x2 γdx 1 dx 2 = 2 x 2 W x 2 1 γdx 1 dx 2 + Γ x2 W x γdx 1 Γ + x W x 2 1 γdx 1. (3.17) Integrating the boundary terms by parts we obtain x 2 W x 2 1 x2 γdx 1 dx 2 = 2 x 2 W x 2 1 γdx 1 dx 2 Γ x1 x2 W x1 γdx Γ + x1 x W x1 γdx 1. (3.18) Applying the Schwarz inequality and using (3.3e), we immediately have x 2 2 W x 2 1 γdx 1 dx 2 W 2 γ 2 Ch p u p+2 W 2. (3.19) The boundary terms in (3.18) are estimated by applying the Schwarz inequality with (2.7) and (3.3d) to obtain Γ 1 x1 x2 W x1 γdx 1 C(h 1/2 W 2 + h 1/2 W 3 )(h 1/2 γ 1 + h 1/2 γ 2 ) Ch p u p+2, ( W 2, + h W 3, ) Ch p u p+2, W 2,. (3.20) We further integrate by parts the boundary terms in (3.14) and use (3.1) to obtain x2 W x 2 1 (u πu)dx 1 = x1 x2 W x1 (u πu)dx 1. (3.21) Γ 1 Γ 1 Since x1 x2 W and x1 (u πu) are continuous along the horizontal edge Γ shared by two neighboring elements i and j, the contribution of such terms to (3.7) can be written as (ā i α,β ā j α,β Γ ) x1 x2 W x1 (u πu)dx 1. (3.22) Applying the Schwarz inequality and using (2.6a), (2.7), (2.8) and the smoothness of the coefficients of L we obtain (ā i α,β ā j α,β Γ ) x1 x2 W x1 (u πu)dx 1 < 9

10 C( W 2 + h W 3 )( u πu 1 + h u πu 2 ) < C W 2,i j u πu 1,i j < Ch p W 2,i j u p+2,i j. (3.23) Following similar steps we prove [ a α,β Dα (u πu)d β ] W dx 1 dx 2 < Ch p u p+2,ω W 2,Ω, α, β. Ω Combining ( ) completes the proof of (3.7). (3.24) In the following lemma we state a superconvergence result. Lemma 6. Let U and u H p+2 be solution of (2.4) and (2.2a), respectively and Πu S N,p interpolate u as described in Lemma 5, then U Πu 2 < Ch p u p+2. (3.25) Proof. Subtracting A(W, πu) from the orthogonality condition leads to UsingLemma5withW complete the proof. A(W, U u) = 0, W S N,p (3.26) A(W, U πu) = A(W, u πu). (3.27) = U πu and the ellipticity of the bilinear form A we In the following theorem we show that the finite element error can be divided into a leading component φ plus a higher-order component θ. Lemma 7. Under the conditions of Lemma 6, there exists ɛ 1 such that with e 2 2 = u Πu ɛ 1 (3.28a) ɛ 1 C(u)h 2p 1. (3.28b) Proof. Adding and subtracting Πu to e yields (3.28a) with ɛ 1 = 2(u Πu, Πu U) 2 + Πu U 2 2. (3.29) Applying the Schwarz inequality and using (2.6a) and (3.25) yields (3.28b). Theorem 1. If u and U are as defined in Lemma 6, then e = u U = Φ + Θ, where the restrictions of Φ and Θ to are φ of (3.3b) and Furthermore, θ = γ + πu U. (3.30a) (3.30b) Φ 2 2 Ch 2p 2 u 2 p+1, (3.30c) Θ 2 2 Ch2p u 2 p+1. (3.30d) Proof. Adding and subtracting πu to e and using (3.3a-3.3d) and (3.25) we prove the theorem. 10

11 4 Even-degree a posteriori error estimation Error estimates of even-degree approximations are computed using interior residuals by solving local Petrov-Galerkin problems for the error. In order to show this we substitute in (2.2a) to obtain where The error is approximated by u = e + U (4.1) A(v, e) = g(v), v H 2 0, (4.2) g(v) = (v, f) A(v, U). (4.3) E = b 1 ψ p+1 (x 1 ) + b 2 ψ p+1 (x 2 ), (4.4) on element and the test function v is selected as where v j (x) = (x j x j,i )δ(x 1 )δ(x 2 ), (4.5) δ(z) = ψ p+1( z,z) z z. (4.6) Since ψ p+1 (x j )andv j (x),j = 1, 2vanishon, the error estimate satisfies the local Dirichlet problems A (v j,e) = g (v j ), j = 1, 2, (4.7) where the subscript denotes that inner products are restricted to. Using (3.30), the finite element problem (4.7) becomes Which in turn can be written as A (v, φ) = g (v) A (v, θ), v H 2 0. (4.8) Ā (v, φ) = g (v) A (v, θ) F (v, φ), v H 2 0, (4.9) where Ā (v, φ) = α = 2 β = 2 a α,β ( x)dα φd β v dx 1 dx 2 (4.10) and F (v, φ) = (a α,β (x) a α,β ( x))dα φd β v dx 1 dx 2 α = 2 β = 2 11

12 + α < 2 β < 2 a α,β (x)dα φd β v dx 1 dx 2. (4.11) Using symmetry properties of ψ p+1 (z) andv j (x) we obtain the uncoupled constantcoefficient problem on element β 1 ā (2,0),(2,0) δ(x 2 )(ψ p+1 (x 1)) 2 dx 1 dx 2 = g(v 1 ) A(v 1,θ) F (v 1,φ), (4.12a) β 2 ā (0,2),(0,2) δ(x 1 )(ψ p+1 (x 2)) 2 dx 1 dx 2 = g(v 2 ) A(v 2,θ) F (v 2,φ). (4.12b) The finite element problem (4.12) may be further simplified by neglecting the higher order terms to obtain the following system for b j,j = 1, 2, b 1 ā (2,0),(2,0) δ(x 2 )(ψ p+1(x 1 )) 2 dx 1 dx 2 = g(v 1 ), (4.13a) b 2 ā (0,2),(0,2) δ(x 1 )(ψ p+1(x 2 )) 2 dx 1 dx 2 = g(v 2 ). (4.13b) The error estimate E in (4.4) with (4.13) is shown to be asymptotically correct in the next theorem. However, prior to this we need the following properties of ψ p+1, δ and v j, j = 1, 2. Lemma 8. Let p>3 be an even integer, then there exit C > 0 and c > 0 such that ch 2(p s)+4 < ψ p+1 2 s, < Ch 2(p s)+4, s = 0, 1, 2, (4.14a) δ(x j )dx 1 dx 2 = Ch p+2, (4.14b) δ(x k )ψ (x j ) 2 dx 1 dx 2 = Ch 3p, j, k = 1, 2, (4.14c) Proof. A direct computation reveals the results. ch 4p < v j 2 2, < Ch4p. (4.14d) In the following lemma we establish bounds for the difference between β j and its approximation b j, j = 1, 2. 12

13 Lemma 9. Let p > 3 be an even integer and u H 2 0 Hp+2. If the coefficients of L in (2.1b) are smooth functions then N (βji 2 b 2 ji) C h u 2 p+2, j = 1, 2. (4.15) Proof. Subtracting (4.13) from (4.12) we obtain (β ji b ji ) 2 = (ā i (2,0),(2,0) (A(v j,θ) + F (v j,φ)) 2 i δ(x 2 )(ψ p+1 (x 1) 2 dx 1 dx 2 ) 2. (4.16) Applying the Schwarz inequality and using (4.14), (3.30d) and the continuity of a α,β (x) wehave A(v j,θ) 2 C v j 2 2, i θ 2 2, i Ch 6p u 2 p+2, i. (4.17) Using the continuity of a α,β (x) with (3.3c) we also prove F (v j,φ) 2 Ch 2 v j 2 2, i φ 2 2, i Ch 6p u 2 p+2, i. (4.18) Combining ( ) with (4.14) yields (β ji b ji ) 2 < C u 2 p+2, i. (4.19) A summation over all elements leads to N (β ji b ji ) 2 < C u 2 p+2. (4.20) Using (3.3b) and (4.14) we obtain βji 2 x 2 = j φ 2 0, i ψ p+1 (x j) 2 < C 0, i h 2 u 2 p+2, i. (4.21) We may solve (4.13) for b j to obtain b 2 ji = g(v j ) 2 (ā i (2,0),(2,0) i δ(x 2 )(ψ p+1 (x 1) 2 dx 1 dx 2 ). (4.22) 2 If we apply the Schwarz inequality, use (4.2) and (4.14), sum over all elements and use (2.6c) we obtain N b 2 ji Ch 2p e 2 2 C h 2 u 2 p+2. (4.23) Combining (4.21) and (4.23) N [b 2 ji + βji] 2 C h 2 u 2 p+2. (4.24) 13

14 Applying the Schwarz inequality [ N N ] 1/2 [ N 1/2 [βji 2 b 2 ji] C (β ji b ji ) 2 (βji 2 + bji)] 2 (4.25) while using (4.20) and (4.24) leads to (4.15). Now we will state and prove the main result of this section. Theorem 2. Let u H 2 0 Hp+2 and U S N,p be solutions of (2.2a) and (2.4), respectively. If p > 3 is an even integer, then where e 2 2 = N 2 j=1 b 2 ji ψ p+1 (x j) 2 0, i + ɛ, (4.26a) ɛ Ch 2p 1 u 2 p+2. (4.26b) Proof. Consider e 2 2 = N [ 2 x1 e 2 0, i + x 2 2 e 2 0, i + x1 x2 e 2 ] 0, i + e 2 1 (4.27) and use (3.30 ) and (3.3b) to obtain N e 2 2 = 2 { [( βji 2 ψ p+1 (x j) 2 ) + x 2 1 θ x 2 1 φ x 2 1 θ i j=1 + 2 x 2 θ x 2 φ 2 x 2 θ + x1 x2 θ 2 ]dx 1 dx 2 } + e 2 1. (4.28) Adding and subtracting b 2 ji ψ p+1 (x j) 2, to the above integrand we obtain (4.26a) with ɛ = N 2 { [( (βji 2 b 2 ji) ψ p+1(x j ) 2 ) + x 2 1 θ x 2 1 φ x 2 1 θ i j=1 + 2 x 2 θ x 2 φ 2 x 2 θ + x1 x2 θ 2 ]dx 1 dx 2 } + e 2 1. (4.29) Applying the Schwarz and triangle inequalities and using the estimates (2.6c), (3.3c - 3.3d), (4.14) and (4.15) yields (4.26b). 14

15 5 Odd-degree a posteriori estimator If u is a smooth solution of (2.1) on, then error estimates E of odd-degree approximations may be computed in terms of jumps in derivatives x 3 j U, j = 1, 2, at the vertices of. Using e = u U we compute the jumps in derivatives at p k = (p 1k,p 2k ),k = 1, 2, 3, 4, of as [ 3 x j e(p k )] j = [ 3 x j U(p k )] j b 1 [ 3 x j ψ p+1 (p 1k )] j + b 2 [ 3 x j ψ p+1 (p 2k )] j j = 1, 2, k = 1, 2, 34, x, (5.1) where [q(p)] j denotes the jump in q at point p in the x j direction. Since jumps in the finite element solution x 3 j U and x 3 j ψ p+1, j = 1, 2, are known, (5.1) yields a linear algebraic system for b 1 and b 2 at each vertex p k. On a rectangular mesh the set of systems (5.1) have a unique solution. More general meshes are treated in [4]. E 2 2, is the average of local errors based on the pairs of b j, j = 1, 2. In the next lemma we state some useful properties of Legendre Polynomials. Lemma 10. Let P p be the Legendre polynomial of degree p, then 1 1 P 2 p (t)dt = 2 2p +1, (5.2a) P p+1 = (2p +1)P p + P p 1, p > 1, (5.2b) P p (+1) = 1, P p ( 1) = ( 1) p, p 0, (5.2c) P p(±1) p+1 p(p +1) = ( 1), 2 p > 0. (5.2d) Proof. See Abramowitz and Stegun [1]. Theorem 3. Let u H 2 0 Hp+2, Πu be as defined in Lemma 5, and p 3 be an odd integer, then u Πu 2 2 = where h 4 64p 2 (p 1) 2 (2p 1) N 2 4 Πu(p k )] 2 j + ɛ 2, (5.3a) j=1 k=1 ɛ 2 Ch 2p 1 u 2 p+2. (5.3b) Proof. The proof will be established in two steps. In the first step we assume u = u j = x p+1 j, j = 1, 2 and show that 2 x j (u j Πu j ) 2 = h 4 64p 2 (p 1) 2 (2p 1) 15 4 Πu j ] 2 j. (5.4) k=1

16 Consider two neighboring elements i and ik and compute the jump in σ j (x j )= x 3 j ψ p+1 = ψ p+1 (x j)inthejth direction at common vertices. On a square-element mesh we have [ 3 x j (u j Πu j )] j = 2σ j (h/2), (5.5) where 2 x j (u j Πu j ) 2 = ψ p+1 (x j) 2 4 k=1 [ 3 x j Πu j ] 2 j 16(σ j (h/2)) 2 = A p 4 Πu j ] 2 j, k=1 (5.6a) A p = ψ p (σ j (h/2)) 2. (5.6b) Transforming the integrals from a physical element to the canonical element, applying the chain rule and using the fact that ψ p+1 P p 1 to obtain Thus, d n dz n ψ p+1(z) = (2/h) n dn dξ n ψ p+1(ξ) = c(2/h) n dn 2 dξ n 2 P p 1(ξ), n = 2, 3. (5.7) A p = h p 2 (p 1) 2 Pp 1(t)dt 2 = 1 h 4 64p 2 (p 1) 2 (2p 1). (5.8) This proves (5.4). Now, we assume w to be a p-degree polynomial with respect to x 1 and x 2.Thus, w Πw = 0, [ xj (w Πw)](p k ) = 0, j = 1, 2. (5.9) Let v = w + β 1 x p β 2 x p+1 2 and i,i= 1, 2, 3, 4 denote, respectively, the right, top, left, and bottom neighbors of with Ω = 4 k=1 k. We define a function v such that v = v 0 and v k = v k such that Πv 0 = Πu, and v 0 (q j ) = u(q j ), (5.10) where q j, j = 1, 2 are two other points in, Πv k = Πu, for x k,k = 1, 2, 3, 4, and (5.11) the following continuity conditions x 3 1 Πv 1 (p + k ) = 3 x 1 Πv 0 (p + k ), k = 1, 2, (5.12a) x 3 2 Πv 2 (p + k ) = 3 x 2 Πv 0 (p + k ), k = 2, 3, (5.12b) 16

17 x 3 1 Πv 3 (p k ) = 3 x 1 Πv 0 (p k ), k = 3, 4, (5.12c) x 3 2 Πv 4 (p k ) = 3 x 2 Πv 0 (p k ), k = 4, 1. (5.12d) The use of the standard aprioriestimate (2.6a) leads to v 0 p+1 u p+1 and (5.13) u v 0 s, Ch p+2 s u p+2,, s = 0, 1, 2. (5.14) Using the continuity conditions (5.12) Πu(p k )] j = Πv 0 (p k )] j, j = 1, 2, k = 1, 2, 3, 4. (5.15) For instance, when j = k = 1, [ x 3 1 Πu(p 1 )] 1 = x 3 1 Πv 1 (p + 1 ) 3 x 1 Πv 0 (p 1 ). (5.16) Using (5.12) we have [ x 3 1 Πu(p 1 )] 1 = [ x 3 1 Πv 0 (p 1 )] 1. (5.17) Adding and subtracting v 0 to u Πu we obtain u Πu 2 2, = u v 0 2 2, + 2(u v 0,v 0 Πu) 2, + v 0 Πu 2 2,. (5.18) Using (5.10) we can write u Πu 2 2, = v 0 Πv 0 2 2, + ɛ, (5.19a) where ɛ = u v 0 2 2, + 2(u v 0,v 0 Πu) 2,. (5.19b) Combining (5.19a), (5.4), (5.15) we obtain 4 2 u Πu 2 2, = A p Πu(p k )] 2 j + ɛ. (5.20) k=1 j=1 Finally, summing over all elements and using (2.6a) we establish (5.3) where ɛ 2 = u v (u v 0,v 0 Πu) 2 (5.21) and ɛ 2 Ch 2p 1 u 2 p+2. (5.22) 17

18 Now we are ready to prove the main theorem of this section. Theorem 4. Let u H 2 0 Hp+2, U S N,p be solutions of (2.2a) and (2.4), respectively. If p 3 is an odd integer, then where u U 2 2 = h 4 64p 2 (p 1) 2 (2p 1) N 2 4 U(p k )] 2 j + ɛ, (5.23a) j=1 k=1 ɛ Ch 2p 1 u 2 p+2. (5.23b) Proof. Adding and subtracting 3 x j Πu(p k ) leads to [ 3 x j U(p k )] 2 j = [ 3 x j Πu(p k )] 2 j + 2[ 3 x j Πu(p k )] j [ 3 x j (U Πu(p k ))] j + [ 3 x j (U Πu(p k ))] 2 j. (5.24) Summing over j, k, multiplying by A p from (5.8) and summing over all elements to obtain where N A p 2 4 N U(p k )] 2 j = A p j=1 k=1 2 4 Πu(p k )] 2 j + ɛ 3, (5.25a) j=1 k=1 ɛ 3 = 2A p N 2 4 Πu(p k )] j (U Πu(p k ))] j + j=1 k=1 N A p 2 4 (U Πu(p k ))] 2 j. (5.25b) j=1 k=1 Let 0 be the canonical element 1 ξ 1,ξ 2 1 and use norm equivalence on the finite dimensional space Q p ( 0 )toshowthat max (ξ1,ξ 2 ) 0 ( 3 ξ 1 v + 3 ξ 2 v ) C v 2,0, v Q p ( 0 ). (5.26) A subsequent linear mapping of 0 to an element i,i= 1, 2,..., N yields max (x1,x 2 ) i ( 3 x 1 w + 3 x 2 w ) C h 2 w 2, i, w Q p ( i ). (5.27) Let i = 4 n=0 i,n, be the union of an element i,0 and its four neighbors i,n, n = 1, 2, 3, 4 having one common edge with it. Using (5.27) [ 3 x j (U Πu)(p k )] j Cmax (x1,x 2 ) i { 3 x 1 (U Πu) + 3 x 2 (U Πu) } 18

19 C h 2 U Πu 2,, j = 1, 2, k = 1, 2, 3, 4. (5.28) i Using (5.28) and (5.8) we obtain 4 N A p k=1 j=1 2 (U Πu)(p k )] 2 j C N U Πu 2 2, i Similarly, using (5.3) we show C U Πu 2 2. (5.29) N A p 2 4 Πu(p k )] 2 j C u Πu 2 2. (5.30) j=1 k=1 Applying the Schwarz inequality we have 2A p N 2 4 Πu(p k )] j (U Πu)(p k )] j j=1 k=1 A p N 2 j=1 k=1 4 Πu(p k )] 2 j 1/2 A p N 2 j=1 k=1 4 (U Πu)(p k )] 2 j 1/2. (5.31) Combining (5.25b),(5.29), (5.30) and (5.31) yields ɛ 3 C U Πu u Πu 2 U Πu 2. (5.32) The estimates (2.6a) and (3.25) imply that ɛ 3 Ch 2p 1 u 2 p+2. (5.33) Combining (3.28), (5.3), (5.25a) and (5.33) yields (5.23) with which completes the proof. ɛ = ɛ 1 + ɛ 2 ɛ 3, (5.34) 6 Error estimators for other finite element spaces The convergence rate is determined by the highest degree polynomial that can be interpolated exactly and the solution smoothness. Thus, piecewise tensorproduct spaces contain many higher order terms that do not increase the convergence rate. Other hierarchical bases [24, 15, 2] lead to better conditioned systems 19

20 and more efficient methods. Hierarchical bases with fewer degrees of freedom for the same order of accuracy are only available for second-order elliptic and parabolic problems. We will construct C 1 hierarchical shape functions and finite element spaces for fourth-order elliptic problems for which the even-odd a posteriori error procedures of Theorem 2 and 4 are still applicable and which provide optimal rates of convergence with fewer degrees than the classical tensor-product shape functions. The terms x p+1 1 and x p+1 2 are the only monomials missing from a bi-p polynomial approximation for it to contain a complete (p + 1)-degree polynomial. These are the only monomial terms needed to make the error estimates of 4 and 5 asymptotically correct when the other monomial terms of degree p + 1arepresent in the solution space S N,p. Theorem 5. Under the conditions of Theorems 2 and 4, let Q p ( i ) be the restriction of S N,p to i (cf. (2.5)) and let M p ( i ) be a space of complete polynomials of degree p on i.ifq p satisfies M p Q p M p+1, M p+1 Q p {x p+1 1, x p+1 2 }, (6.1) then the error estimates (4.26) or (5.23) apply when p is even or odd, respectively. Proof. The proof goes along the same lines as that of Theorems 4 and 2. The conditions (6.1) may be used to construct such finite element spaces for which the error estimates (4.26) and (5.23) are asymptotically correct. However, for efficiency reasons we will use the smallest finite element spaces that satisfy (6.1) Q 3 = {N 0,0 ij,n0,1 ij,n1,0 ij,n1,1 ij, i,j = 1, 1}, (6.2a) Q 4 = Q 3 {N j,4 i,1,nj,4 i,2,nj,4 i,3,nj,4 i,4,, 1, j =0, 1}, (6.2b) Q 5 = Q 4 {N j,5 i,1, Nj,5 i,2, Nj,5 i,3,nj,5 i,4, i =1, 1, j =0, 1}, (6.2c) Q 6 = Q 5 {N j,6 i,1, Nj,6 i,2,nj,6 i,3,nj,6 i,4 4,4,, 1, j =0, 1} N0. (6.2d) As described above, the set Q 3 is the standard bi-cubic C 1 approximation. However, Q 4, Q 5, and Q 6 are smaller than bi p, p =4, 5, 6 spaces with only one interior mode in Q 6. Furthermore, we note that with increasing p, thesetq p contains little more than half the terms in the bi p approximation space. 7 Computational examples We illustrate the performance of the odd-even error estimators for both bi p and hierarchical approximations by solving a fourth-order boundary-value problem using square and rectangular meshes. We will show that the effectivity index η = E 2 / e 2 (7.1) 20

21 Table 1: Errors and effectivity indices for Example 1 on N-element uniform meshes with piecewise bi p polynomial approximations on square mesh. p 3 4 N e 2 η e 2 η (+1) (00) (+0) ( 1) ( 1) ( 2) ( 1) ( 3) p 5 6 N e 2 η e 2 η ( 1) ( 2) ( 3) ( 4) ( 4) ( 5) ( 5) ( 7) approaches unity under mesh refinement. Example 1. We solve the fourth-order boundary-value problem 4 u(x, y) x u(x, y) x 2 y u(x, y) y 4 + u(x, y) = f(x, y), (x, y) Ω with the boundary conditions (7.2a) u(x, y) u(x, y) = 0, = 0, (x, y) Ω. (7.2b) ν We select Ω = [0, 1] [0, 1] and the right hand side f(x, y) such that the exact solution is given by u(x, y) = sin(πx) 2 sin(πy) 2. (7.2c) We solve (7.2) on uniform square-elements having 16, 64, 256, and 1024 elements using bi p approximations of orders 3 to 6. Errors and global effectivity indices in the H 2 norm are presented in Table 1. The a posteriori estimators behave as predicted by the theory and converge to the true errors under h refinement. Results of Table 1, as reported in [4], indicate that even-degree estimators perform better that odd-degree estimators. Example 2 We solve (7.2) on Ω = [0, 2] [0, 1] on rectangular meshes having 16, 64, 256, and 1024 elements using hierarchical bases defined in (6.2) with degrees from 3 to 6. We present the true errors and effectivity indices in the H 2 norm in Table 2. Although we did not do the analysis for rectangular-element meshes, the results in Table 2 show that estimators converge to the true error. The global effectivity indices converge to unity under mesh refinement for the smaller sets Q p of (6.2). Again, even-degree estimators perform better than odd-degree estimators. 21

22 Table 2: Errors and effectivity indices for Example 2 on N-element uniform meshes with piecewise hierarchical polynomial approximations on a rectangular mesh. p 3 4 N e 2 η e 2 η (+1) (+1) (+1) (+0) ( 1) ( 1) ( 1) ( 2) p 5 6 N e 2 η e 2 η ( 1) ( 1) ( 1) ( 2) ( 3) ( 4) ( 4) ( 5) Conclusion We have developed asymptotically exact a posteriori error estimates for twodimensional fourth-order boundary value problems based on the even-odd dichotomy of Babuska and Yu with tensor product bi p finite element approximations. Although, both even- and odd-degree a posteriori error estimators are computationally simple, the even-degree estimator does not involve neighbors communication which is more efficient for parallel implementation and appears to converge to the true error faster as the mesh is h-refined. Finally, we showed that both even- and odd-degree estimators converge to the true error using a more general and efficient hierarchical finite element approximations. Extension of the analysis to hexahedral meshes and higher-order elliptic equations is straight forward. Convergence analyses for nonlinear problems and more general meshes including unstructured quadrilateral, triangular and tetrahedral meshes are less obvious and will be investigated in the future. Acknowledgment This Research was supported by the National Science Foundation (Grant Number ASC ). References [1] M. Abramowitz and I.A. Stegun, editors. Handbook of Mathematical Functions. Dover, New York,

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