A C 0 -Weak Galerkin Finite Element Method for the Biharmonic Equation

Transcription

1 J Sci Comput (2014) 59: DOI /s A C 0 -Weak Galerkin Finite Element Method for the Biharmonic Equation Lin Mu Junping Wang Xiu Ye Shangyou Zhang Received: 3 December 2012 / Revised: 9 June 2013 / Accepted: 13 August 2013 / Published online: 24 August 2013 Springer Science+Business Media New York 2013 Abstract A C 0 -weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for C 0 functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete H 2 norm and the standard H 1 and L 2 norms with appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott- Zhang interpolation operator is constructed to assist the corresponding error estimates. This refined interpolation preserves the volume mass of order (k + 1 d) and the surface mass of order (k + 2 d) for the P k+2 finite element functions in d-dimensional space. Keywords Weak Galerkin Finite element methods Weak Laplacian Biharmonic equation Triangular mesh Tetrahedral mesh Junping Wang The research of Wang was supported by the NSF IR/D program, while working at National Science Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation, Xiu Ye This research was supported in part by National Science Foundation Grant DMS L. Mu Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA linmu@math.msu.edu J. Wang Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230, USA jwang@nsf.gov X. Ye Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA xxye@ualr.edu S. Zhang (B) Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA szhang@udel.edu

2 474 J Sci Comput (2014) 59: Mathematics Subject Classification Primary 65N30 65N15 Secondary 35B45 1 Introduction In this paper, we are concerned with robust numerical methods for the biharmonic equation of the following form 2 u = f, in, (1) u = g, on, (2) u = φ, n on, (3) where is a bounded polygonal or polyhedral domain in R d for d = 2, 3. For the biharmonic problem (1) with Dirichlet and Neumann boundary conditions (2) and(3), the corresponding variational form is given by seeking u H 2 ( ) satisfying u = g and u n = φ such that ( u, v)= ( f,v) v H0 2 ( ), (4) where H0 2( ) is the subspace of H 2 ( ) consisting of functions with vanishing value and normal derivative on. Conforming finite element methods for the forth order problem (4) require the finite element space be a subspace of H 2 ( ). It is well known that constructing H 2 conforming finite elements is generally quite challenging, especially in three and higher dimensional spaces. The weak Galerkin finite element method, first introduced in [28](seealso[27]and [22]), uses discontinuous finite element functions to relax the difficulty in the construction of conforming elements. Unlike the classical nonconforming finite element method where standard derivatives are taken on each element, the weak Galerkin finite element method relies on weak derivatives taken as approximate distributions for functions in nonconforming finite element spaces. In general, weak Galerkin methods refer to finite element techniques for partial differential equations in which differential operators (e.g., gradient, divergence, curl, Laplacian) are approximated by weak forms as distributions. A weak Galerkin finite element method for the biharmonic equation has been derived in [23] by using totally discontinuous functions of piecewise polynomials on general partitions of arbitrary shape of polygons/polyhedra. The key of the method in [23] lies in the use of a discrete weak Laplacian plus a stabilization that is parameter-free. In this paper, we will follow the approach of [23] and shall develop a new weak Galerkin method for the biharmonic equation by using C 0 finite element functions. Comparing with the WG method developed in [23], the C 0 -weak Galerkin finite element scheme has less number of unknowns due to the continuity requirement. On the other hand, due to the very same continuity requirement, the C 0 -WG method allows only traditional finite element partitions (such as triangles/quadrilaterals in 2D), instead of arbitrary polygonal/polyhedral grids as allowed in [23]. A suitably-designed interpolation operator is needed for the convergence analysis of the C 0 -weak Galerkin method. This paper shall introduce a refined version of the Scott-Zhang operator [26] which preserves the volume mass up to order (k + 1 d), andthesurface mass up to order (k + 2 d), when interpolating H 1 functions to the P k+2 C 0 -finite element space; namely, if Q 0 : H 1 ( ) C 0 -P k+2 is the interpolation operator into the space of continuous piecewise polynomial of degree k + 2, then

3 J Sci Comput (2014) 59: (v Q 0 v)pdt = 0 p P k+1 d (T ), T (v Q 0 v)pde = 0 p P k+2 d (T ), E where T is any triangle (d = 2) or tetrahedron (d = 3) in the finite element, and E is an edge or a face-triangle of T. We note that a pointwise version of this interpolation operator (only for 2D H 2 functions) was constructed by Girault and Raviart, Page 100 in [14]. With the operator Q 0, we can show an optimal order of approximation property of the C 0 -finite element space, under the constraints of weak Galerkin formulation. Consequently, we show optimal order of convergence in both a discrete H 2 norm and the standard H 1 and L 2 norms, for the C 0 weak Galerkin finite element solution. The biharmonic equation models a plate bending problem, which is one of the first applicable problems of the finite element method, cf. [2,8,13,33]. The standard finite element method, i.e., the conforming element, requires a C 1 function space of piecewise polynomials. This would lead to a high polynomial degree [2,29,31,32], or a macro-element [8,10,13,16,25,30], or a constraint element (where the polynomial degree is reduced at inter-element boundary) [4,24,33]. Mixed methods for the biharmonic equation avoid the use of C 1 functions by reducing the fourth order equation to a system of two second order equations, [3,12,15,18,21]. Many other different nonconforming and discontinuous finite element methods have been developed for solving the biharmonic equation. Morley element [19] is a well known nonconforming element for its simplicity. C 0 interior penalty methods are studied in [7,11,20], which share some similarity to our C 0 -weak Galerkin method. But our weak Galerkin formulation is symmetric, positive definite, and absolutely stable. 2 Weak Laplacian and Discrete Weak Laplacian Let D be a bounded polyhedral domain in R d, d = 2, 3. We use the standard definition for the Sobolev space H s (D) and their associated inner products (, ) s,d, norms s,d,and seminorms s,d for any s 0. When D =, we shall drop the subscript D in the norm and in the inner product. Let T be a triangle or a tetrahedron with boundary T.Aweak function on the region T refers to a function v ={v 0,v b, v n } such that v 0 L 2 (T ), v b H 1 2 ( T ), andv n n H 1 2 ( T ),wheren is the outward normal direction of T on its boundary. The first and second components v 0 and v b can be understood as the value of v in the interior and on the boundary of T. The third component v n intends to represent v on the boundary of T. Note that v b and v n may not be necessarily related to the trace of v 0 and v 0 on T, respectively. Denote by W (T ) the space of all weak functions on T ; i.e., W (T ) ={v ={v 0,v b, v n }: v 0 L 2 (T ), v b H 1 2 ( T ), vn n H 1 2 ( T )}. (5) Let, T be the inner product in L 2 ( T ). DefineG 2 (T ) by G 2 (T ) ={ϕ : ϕ H 1 (T ), ϕ L 2 (T )}. For any ϕ G 2 (T ), wehave ϕ H(div, T ), and hence ϕ n H 1 2 ( T ). Definition 2.1 The dual of L 2 (T ) can be identified with itself by using the standard L 2 inner product as the action of linear functionals. With a similar interpretation, for any v W (T ),

4 476 J Sci Comput (2014) 59: the weak Laplacian of v ={v 0,v b, v n } is defined as a linear functional w v in the dual space of G 2 (T ) whose action on each ϕ G 2 (T ) is given by ( w v, ϕ) T = (v 0, ϕ) T v b, ϕ n T + v n n,ϕ T, (6) where n is the outward normal direction to T. The Sobolev space H 2 (T ) can be embedded into the space W (T ) by an inclusion map i W : H 2 (T ) W (T ) defined as follows i W (φ) ={φ T,φ T, φ T }, φ H 2 (T ). With the help of the inclusion map i W, the Sobolev space H 2 (T ) can be viewed as a subspace of W (T ) by identifying each φ H 2 (T ) with i W (φ). Analogously, a weak function v = {v 0,v b, v n } W (T ) is said to be in H 2 (T ) if it can be identified with a function φ H 2 (T ) through the above inclusion map. It is not hard to see that the weak Laplacian is identical with the strong Laplacian in H 2 (T ); i.e., w v = v for all functions v H 2 (T ). Next, we introduce a discrete weak Laplacian operator by approximating w in a polynomial subspace of the dual of G 2 (T ). To this end, for any non-negative integer r 0, denote by P r (T ) the set of polynomials on T with degree no more than r. A discrete weak Laplacian operator, denoted by w,r,t, is defined as the unique polynomial w,r,t v P r (T ) satisfying the following equation ( w,r,t v, ϕ) T = (v 0, ϕ) T v b, ϕ n T + v n n,ϕ T, ϕ P r (T ). (7) Since the quantity of interest is not v n but v n n, we shall replace v n by v n n with v n a scalar so that v n n = v n n n. Thus, it is clear that v n = v n n. By doing so, one may reduce the number of unknowns in the formulation of the corresponding finite element method. Note that the scalar v n represents v n if v n = v. 3 Weak Galerkin Finite Element Methods Let T h be a triangular (d = 2) or a tetrahedral (d = 3) partition of the domain with mesh size h. Denote by E h the set of all edges or faces in T h,andlete 0 h = E h\ be the set of all interior edges or faces. Obviously, v n in v n n is dependent on n. To ensure v n a single valued function on e E h, we introduce a set of normal directions on E h as follows D h ={n e : n e is unit and normal to e, e E h }. (8) Since C 0 weak Galerkin element is considered, the weak function v ={v 0,v b,v n n e } defined in (5) can be written as v ={v 0,v 0,v n n e } where v b is the trace of v 0 on T. Then, we can define a weak Galerkin finite element space V h for k 0 as follows where V h ={v ={v 0,v 0,v n n e }: v 0 V 0,v n e P k+1 (e), e T } (9) V 0 ={v H 1 ( ); v T P k+2 (T )}. (10) For simplicity, we will replace weak function v ={v 0,v 0,v n n e } by v ={v 0,v n n e } from now on.

5 J Sci Comput (2014) 59: Denote by V 0 h a subspace of V h with vanishing traces; i.e., Vh 0 ={v ={v 0,v n n e } V h,v 0 e = 0, v n e = 0, e T }. Denote by h the trace of V h on from the component v 0. It is easy to see that h consists of piecewise polynomials of degree k + 2. Similarly, denote by ϒ h the trace of V h from the component v n as piecewise polynomials of degree k + 1. Let w,k be the discrete weak Laplacian operator on the finite element space V h computed by using (7) on each element T for k 0; i.e., ( w,k v) T = w,k,t (v T ) v V h. (11) For simplicity of notation, from now on we shall drop the subscript k in the notation w,k for the discrete weak Laplacian. We also introduce the following notation ( w v, w w) h = ( w v, w w) T. For any u h ={u 0, u n n e } and v ={v 0,v n n e } in V h, we introduce a bilinear form as follows s(u h,v)= T u 0 n e u n, v 0 n e v n T. (12) h 1 The stabilizer s(u h,v)defined above is to enforce a connection between the normal derivative of u 0 along n e and its approximation u n. Weak Galerkin Algorithm 1 A numerical approximation for (1)-(3) can be obtained by seeking u h ={u 0, u n n e } V h satisfying u 0 = Q b g and u n = (n n e )Q n φ on and the following equation: ( w u h, w v) h + s(u h,v)=( f, v 0 ) v ={v 0,v n n e } Vh 0, (13) where Q b g and Q n φ are the standard L 2 projections onto the trace spaces h and ϒ h, respectively. Lemma 3.1 The weak Galerkin finite element scheme (13) has a unique solution. Proof It suffices to show that the solution of (13) istrivialif f = g = φ = 0. To this end, assume f = g = φ = 0andtakev = u h in (13). It follows that ( w u h, w u h ) h + s(u h, u h ) = 0, which implies that w u h = 0 on each element T and u 0 n e = u n on T. We claim that u h = 0 holds true locally on each element T. To this end, it follows from w u h = 0and (7) that for any ϕ P k (T ) we have 0 = ( w u h,ϕ) T = (u 0, ϕ) T u 0, ϕ n T + u n n e n, ϕ T (14) = ( u 0,ϕ) T + u n n e n u 0 n, ϕ T = ( u 0,ϕ) T, wherewehaveused u n n e n u 0 n =±(u n u 0 n e ) = 0 (15) in the last equality. The identity (14) implies that u 0 = 0 holds true locally on each element T. This, together with u 0 n e = u n on T, shows that u h is a smooth harmonic function globally on. The boundary condition of u 0 = 0andu n = 0 then implies that u h 0on, which completes the proof.

6 478 J Sci Comput (2014) 59: Projections: Definition and Approximation Properties In this section, we will introduce some locally defined projection operators corresponding to the finite element space V h with optimal convergent rates. Let Q 0 : H 1 ( ) V 0 be a special Scott-Zhang interpolation operator, to be defined in (54) in Appendix, such that for given v H 1 ( ), Q 0 v V 0 and for any T T h, (Q 0 v, ϕ) T Q 0 v, ϕ n T = (v, ϕ) T v, ϕ n T, ϕ P k (T ), (16) and for 0 s 2 h 2s u Q 0 u 2 s,t 1/2 Ch k+3 u k+3. (17) Now we can define an interpolation operator Q h from H 2 ( ) to the finite element space V h such that on the element T,wehave Q h u ={Q 0 u,(q n ( u n e ))n e }, (18) where Q 0 is defined in (54) andq n is the L 2 projection onto P k+1 (e), for each e T.In addition, let Q h be the local L 2 projection onto P k (T ). Foranyϕ P k (T ) we have ( w Q h u,ϕ) T = (Q 0 u, ϕ) T Q 0 u, ϕ n T + Q n ( u n e )n e n, ϕ T which implies = (u, ϕ) T u, ϕ n T + u n, ϕ T = ( u, ϕ) T = (Q h u, ϕ) T, w Q h u = Q h ( u). (19) The above identity indicates that the discrete weak Laplacian of a projection of u is a good approximation of the Laplacian of u. Let T T h be an element with e as an edge or a face triangle. It is well known that there exists a constant C such that for any function g H 1 (T ) ( ) g 2 e C h 1 g 2 T + h T g 2 T. (20) T Define a mesh-dependent semi-norm in the finite element space V h as follows v 2 = ( w v, w v) h + s(v, v), v V h. (21) Using (20), we can derive the following estimates which are useful in the convergence analysis for the WG-FEM (13). Lemma 4.1 Let w H k+3 ( ) and v V h. Then there exists a constant C such that the following estimates hold true. w Q h w, ( v 0 v n n e ) n T Ch k+1 w k+3 v, (22) h 1 T T T ( Q 0w) n e Q n ( w n e ), v 0 n e v n T (23) h Ch k+1 w k+3 v.

7 J Sci Comput (2014) 59: Proof To derive (22), we can use the Cauchy-Schwarz inequality, (15), the trace inequality (20), and the definition of Q h to obtain w Q h w, ( v 0 v n n e ) n T h T w Q h w 2 2 T h 1 T v 0 n e v n 2 T 1 C 2 ( w Qh w 2 T + h2 T ( w Q h w) 2 ) T v Ch k+1 w k+3 v. As to (23), we have from the definition of Q n, the Cauchy-Schwarz inequality, the trace inequality (20), and (17)that h 1 T ( Q 0w) n e Q n ( w n e ), v 0 n e v n T = T ( Q 0w) n e w n e, v 0 n e v n T h 1 h 1 T C Ch k+1 w k+3 v. This completes the proof. 1 2 ( Q 0w w) n e 2 T h 1 T ( h 2 T Q 0w w 2 T + Q 0w w 2 1,T v 0 n e v n 2 T ) 1 2 v An Error Equation We first derive an equation that the projection of the exact solution, Q h u, shall satisfy. Using (7), the integration by parts, and (19), we obtain ( w Q h u, w v) T = (v 0, ( w Q h u)) T + (v n n e ) n, w Q h u T v 0, ( w Q h u) n T = ( v 0, w Q h u) T + v 0, ( w Q h u) n T v 0 n, w Q h u T + (v n n e ) n, w Q h u T v 0, ( w Q h u) n T = ( v 0, w Q h u) T ( v 0 v n n e ) n, w Q h u T = ( v 0, Q h u) T ( v 0 v n n e ) n, Q h u T = ( u, v 0 ) T ( v 0 v n n e ) n, Q h u T, which implies that ( u, v 0 ) T = ( w Q h u, w v) T + ( v 0 v n n e ) n, Q h u T. (24)

8 480 J Sci Comput (2014) 59: Next, it follows from the integration by parts that ( u, v 0 ) T = ( 2 u,v 0 ) T + u, v 0 n T ( u) n,v 0 T. Summing over all T and then using the identity ( 2 u,v 0 ) = ( f,v 0 ) we arrive at ( u, v 0 ) T = ( f,v 0 ) + u, v 0 n T = ( f,v 0 ) + u,( v 0 v n n e ) n T. Combining the above equation with (24) leads to ( w Q h u, w v) h = ( f,v 0 ) + u Q h u,( v 0 v n n e ) n T. (25) Define the error between the finite element approximation u h and the projection of the exact solution u as e h := {e 0, e n n e }={Q 0 u u 0,(Q n ( u n e ) u n )n e }. Taking the difference of (25)and(13) gives the following error equation ( w e h, w v) h + s(e h,v) = u Q h u,( v 0 v n n e ) n T +s(q h u,v) v V 0 h. (26) Observe that the definition of the stabilization term s(, ) indicates that s(q h u,v)= T ( Q 0u) n e Q n ( u n e ), v 0 n e v n T. h 1 6 Error Estimates First, we derive an estimate for the error function e h in the natural triple-bar norm, which can be viewed as a discrete H 2 -norm. Theorem 6.1 Let u h V h be the weak Galerkin finite element solution arising from (13) with finite element functions of order k Assume that the exact solution of (1) (3) is regular such that u H k+3 ( ). Then, there exists a constant C such that u h Q h u Ch k+1 u k+3. (27) Proof By letting v = e h in the error Eq. (26), we obtain the following identity e h 2 = u Q h u,( e 0 e n n e ) n T + T ( Q 0u n e Q n ( u n e ), e 0 n e e n T. h 1 Using the estimates of Lemma 4.1, we arrive at e h 2 Ch k+1 u k+3 e h, which implies (27). This completes the proof of the theorem.

9 J Sci Comput (2014) 59: Wenotethat,sincealltheerrorestimatesarelocal,wehavealocalversionof(27), u h Q h u C T 1/2 u 2 H s (T ), s 2. h 2(s 2) Next, we would like to establish some error estimates for the first component of the error function e h in the standard L 2 and H 1 norms. To this end, we consider the following dual problem 2 w = φ in, (28) w = 0, on, (29) w n = 0 on. (30) Assume that the dual problem has the usual H r regularity in the sense that there exists a constant C such that for r=3,4 w r C φ r 4. (31) For smooth domains, the regularity H 2+ι can be proved for φ H ι 2 ( ) and for any ι>0, in any dimension [9]. In 2D, a regularity of H 4 has been obtained on convex polygonal domains with all inner angles smaller than by Blum and Rannacher [5]. An H 5/2 regularity was obtained on n-dimensional Lipschitz domains in [1]. Theorem 6.2 Let u h V h be the weak Galerkin finite element solution arising from (13). Assume that the exact solution of (1) (3) is regular such that u H k+3 ( ) and (31) holds. Then, there exists a constant C such that for k > 0 for k = 0 and for k 0 Q 0 u u 0 Ch k+3 u k+3, (32) Q 0 u u 0 Ch 2 u 3, (33) (Q 0 u u 0 ) Ch k+2 u k+3. (34) Proof Let w be the solution of (28) (30). Using the integration by parts gives ( 2 w, e 0 ) = ( w, e 0 ) T w, e 0 n T = ( w, e 0 ) T w, ( e 0 e n n e ) n T. Using (24) with w in the place of u, we can rewrite the above equation as follows ( 2 w, e 0 ) = ( w Q h w, w e h ) h w Q h w, ( e 0 e n n e ) n T. It now follows from the error Eq. (26)that ( w Q h w, w e h ) h = u Q h u,( Q 0 w Q n ( w n e )n e ) n T s(e h, Q h w) + s(q h u, Q h w).

10 482 J Sci Comput (2014) 59: Combining the two equations above gives ( 2 w, e 0 ) = w Q h w, ( e 0 e n n e ) n T (35) + u Q h u,( Q 0 w Q n ( w n e )n e ) n T s(e h, Q h w) + s(q h u, Q h w). Using the estimates of Lemma 4.1, we can bound two terms on the right-hand side of the equation above as follows, { Ch w 3 e w Q h w, ( e 0 e n n e ) n T h, k 0 Ch 2 w 4 e h, k > 0. { Ch w 3 e s(e h, Q h w) h, k 0 Ch 2 w 4 e h, k > 0. It follows from (15) and the definition of Q n and Q 0 that ( Q 0 w Q n ( w n e )n e ) n e T (36) = Q 0 w n Q n ( w n) T Q 0 w n w n T + w n Q n ( w n) T 2 Q 0 w n w n T. Using (36)and(20), we have, u Q h u,( Q 0 w Q n ( w n e )n e ) n T C 1/2 h u Q h u 2 T h 1 ( Q 0 w w) n 2 T C 1/2 ( u Qh u 2 T + h2 T ( u Q h u) 2 ) T ( h 2 Q 0 w w 2 T + ( Q 0w w) 2 ) T { Ch k+2 u k+3 w 3, k 0 Ch k+3 u k+3 w 4, k > 0. Using (36)and(20), we have s(q h u, Q h w) = 1 h (Q 0u) n e Q n ( u n e ), (Q 0 w) n e Q n ( w n e ) T 1 h ( Q 0u u) n 2 T { Ch k+2 u k+3 w 3, k 0 Ch k+3 u k+3 w 4, k > 0. 1/2 1/2 1/2 1 h ( Q 0w w) n 2 T 1/2

11 J Sci Comput (2014) 59: Substituting all above estimates into (35) and using (27) give that for k 0 and for k > 0 ( 2 w, e 0 ) Ch k+2 u k+3 w 3, (37) ( 2 w, e 0 ) Ch k+3 u k+3 w 4. (38) Let φ = e 0 in (28). Testing (28) byerrorfunctione 0 and using (37) and(38), we have that for k > 0 and for k = 0 e 0 2 = ( 2 w, e 0 ) Ch k+3 u k+3 w 4, e 0 2 = ( 2 w, e 0 ) Ch 2 u 3 w 3 Ch 2 u 3 w 4. Combining the above two estimates and (31) with r = 4andφ = e 0, we obtain the L 2 error estimates (32)and(33). To derive (34), let φ = q H 1 ( ) in (28) whereq = e 0 is a vector-valued L 2 function. Then testing (28) by the error function e 0 gives that for k 0 ( 2 w, e 0 ) = ( q, e 0 ) = ( e 0, e 0 ). (39) With an H 3 regularity for the dual problem, we have from (31)that Using (39), (37)and(40), we have that for k 0 w 3 C q 1 C e 0 1. (40) (Q 0 u u 0 ) =( 2 w, e 0 ) Ch k+2 u k+3, which proves (34). The L 2 error estimate in Theorem 6.2 should be sharp. Hu and Shi [17] proved an O(h 2 ) lower bound for all known conforming and almost all non-conforming finite elements of P 2 or Q 2 type, in L 2 norm. We believe that similar results can be derived for WG finite element solutions with k = 0. Interested readers are encouraged to conduct a thorough study of this issue. The rest of this section is devoted to a study of the error in H 2 norm. To this end, we define a discrete semi-h 2 norm as follows v 2,h = i, j=1 d i j v 2 0,T Corollary 6.1 Let u h ={u 0, u n n e } V h be the weak Galerkin finite element solution arising from (13). Assume that the exact solution of (1) (3) is regular such that u H k+3 ( ) for k 0 and that (31) holds true with r = 3. Then, there exists a constant C such that 1 2 u u 0 2,h Ch k+1 u k+3, (41) u u 0 1 Ch k+2 u k+3. (42).

12 484 J Sci Comput (2014) 59: Proof Let Q 0 u be the Scott-Zhang interpolation of u defined in (16). By the triangle inequality, the inverse inequality, and the optimal order H 1 error estimate (34), we obtain u u 0 2,h u Q 0 u 2,h + Q 0 u u 0 2,h Ch k+1 u k+3 + Ch 1 Q 0 u u 0 1 Ch k+1 u k+3. Analogously, we can prove (42) by using a similar triangle inequality. 7 Numerical Experiments This section shall report some numerical results for the C 0 weak Galerkin finite element methods for the following biharmonic equation: 2 u = f in, (43) u = g on, (44) u = ψ n on. (45) For simplicity, all the numerical experiments are conducted by using k = 0ork = 1inthe finite element space V h in (9). If φ P 0 (T ) (i.e. k = 0), the Eq. (7) can be simplified as ( w v, φ) T = v n n e n, φ T. The error for the C 0 -WG solution will be measured in four norms defined as follows: H 1 semi-norm: v v = v v 0 2 dx. T Discrete H 2 norm: v 2 = w v 2 T + h 1 v 0 n e v n 2 T, Element-based L 2 norm: Q 0 v v 0 2 = Q 0 v v 0 2 dx, T Edge-based L 2 norm: Q n ( v n e ) v n 2 b = h Q n ( v n e ) v n 2 ds. e E h 7.1 Example 1 Consider the biharmonic problem (43) (45) in the square domain = (0, 1) 2. Set the exact solution by u = x 2 (1 x) 2 y 2 (1 y) 2. e

13 J Sci Comput (2014) 59: Table 1 Example 1. Convergence rate for element P 2 (T ) P 1 (e) (k = 0) h u u 0 1 u h Q h u u 0 Q 0 u Q n ( u n e ) u n b 1/ e e e e 03 1/ e e e e 03 1/ e e e e 04 1/ e e e e 04 1/ e e e e 05 1/ e e e e 06 Order Table 2 Example 1. Convergence rate for element P 3 (T ) P 2 (e) (k = 1) h u u 0 1 u h Q h u u 0 Q 0 u Q n ( u n e ) u n b 1/ e e e e 03 1/ e e e e 04 1/ e e e e 05 1/ e e e e 06 1/ e e e e 07 1/ e e e e 08 Order It is easy to check that u = 0, u n = 0. The function f is given according to the Eq. (43). The test is performed by using uniform triangular mesh. The mesh is constructed as follows: 1) partition the domain into n n sub-rectangles; 2) divide each square element into two triangles by the diagonal line with a negative slope. The mesh size is denoted by h = 1/n. Table 1 shows the convergence rate for C 0 -WG solutions based on k = 0in four norms respectively. The numerical results indicate that the WG solution is convergent with rate O(h 2 ) in H 1, O(h 1 ) in H 2,andO(h 2 ) in L 2 norms. The convergence rate for Q n ( u n e ) u n b is O(h 2 ). Also, the same problem is tested for k = 1. The results are reported in Table 2. It indicates that the WG solution is convergent with rate O(h 3 ) in H 1, O(h 2 ) in H 2,andO(h 4 ) in L 2 norms. We note that the L 2 error is convergent at order 4, two orders higher than that of k = 0, confirming the sharpness of Theorem 6.2. Moreover the convergence rate for Q n ( u n e ) u n b is O(h 3 ),fork = Example 2 In this problem, we set = (0, 1) 2 and the exact solution: u = sin(π x) sin(πy),

14 486 J Sci Comput (2014) 59: Table 3 Example 2. Convergence rate for element P 2 (T ) P 1 (e) (k = 0) h u u 0 1 u h Q h u u 0 Q 0 u Q n ( u n e ) u n b 1/ e e e 01 1/ e e e 02 1/ e e e 02 1/ e e e e 03 1/ e e e e 03 1/ e e e e 04 Order Table 4 Example 3. Convergence rate for element P 2 (T ) P 1 (e) (k = 0) h u u 0 1 u h Q h u u 0 Q 0 u Q n ( u n e ) u n b 1/ e e e 01 1/ e e e 01 1/ e e e 02 1/ e e e e 02 1/ e e e e 03 1/ e e e e 04 Order with u = 0, u = 0. n Boundary conditions and f are given according to the Eqs. (43) (45). Again, the uniform triangular mesh is used in the experiment. Table 3 shows that the convergence rate for C 0 -WG solutions in H 1, H 2 and L 2 norms is O(h 2 ), O(h) and O(h 2 ), respectively. 7.3 Example 3 The exact solution is chosen as u = sin(π x) cos(πy), with nonhomogeneous boundary conditions. Table 4 shows that the convergence rate for C 0 -WG solutions in H 1, H 2 and L 2 norms is O(h 2 ), O(h), ando(h 2 ), respectively. 7.4 Example 4 In the final example, we test the a case where the exact solution has a low regularity in the domain = (0, 1) 2. The exact solution is given by ( u = r 3/2 sin 3θ 2 3sin θ ), 2

15 J Sci Comput (2014) 59: Table 5 Example 4. Convergence rate for element P 2 (T ) P 1 (e)(k = 0) h u u 0 1 u h Q h u u 0 Q 0 u Q n ( u n e ) u n b 1/ e e e e 01 1/ e e e e 02 1/ e e e e 02 1/ e e e e 03 1/ e e e e 03 1/ e e e e 03 Order Table 6 Example 4. Convergence rate for element P 3 (T ) P 2 (e) (k = 1) h u u 0 1 u h Q h u u 0 Q 0 u Q n ( u n e ) u n b 1/ e e e e 02 1/ e e e e 02 1/ e e e e 03 1/ e e e e 03 1/ e e e e 04 1/ e e e e 04 Conv.Rate e where (r,θ)are the polar coordinates. It is known that u H 2.5 ( ). The performance for C 0 weak Galerkin finite element approximations for element P 2 (T ) P 1 (e)(k = 0) is reported in Table 5. The convergence rates in H 1 -norm, H 2 norm, and edge-based L 2 -norm are seen as O(h 1.4 ), O(h 0.47 ),ando(h 1.4 ). The corresponding theoretical prediction has the order of O(h 1.5 ), O(h 0.5 ),ando(h 1.5 ). We believe that the numerical results are in consistency with the theory. Table 5 indicates that the numerical convergence rate in the standard L 2 is of order O(h 1.88 ), which exceeds the theoretical prediction of O(h 1.5 ). Table 6 contains some numerical results for the weak Galerkin element P 3 (T ) P 2 (e) (k = 1). The convergence rates in H 1 -norm, H 2 norm, and edge-based L 2 -norm are seen as O(h 1.5 ), O(h 0.5 ),ando(h 1.5 ), which are completely in consistency with the theory. For the element-based L 2 error, Table 6 indicates a numerical convergence rate of order O(h 2.49 ), which is also consistent with the theoretical prediction of O(h 2.5 ). 8 Appendix: A Mass-Preserving Scott-Zhang Operator We will prove the existence of an interpolation Q 0 used in (16) and in the previous section, which is a special Scott-Zhang operator [26]. The new Scott-Zhang operator preserves the mass on each element and on each face, of four orders and three orders less, respectively, when interpolating H 1 ( ) functions to the finite element V h functions. We shall derive the optimal-order approximation properties for the interpolation in the section, which leads to a quasi-optimal convergence of the weak Galerkin finite element method (13). The original Scott-Zhang operator maps u H 1 ( ) functions to C 0 -Lagrange finite element functions, preserving the zero boundary condition if u H 1 ( ). It is an Lagrange interpolation. All the Lagrange nodes ([6]) on one element are classified into three types:

16 488 J Sci Comput (2014) 59: Fig. 1 Averaging patches (C j ) in 2D (an edge or a triangle), for corner nodes (c j ), middle nodes (m j )and internal nodes (i j ), in 2D Fig. 2 Averaging patches (C j ) in 3D (a triangle or a tetrahedron), for corner nodes (c j ), middle nodes (m j ) and internal nodes (i j ),in3d corner nodes c j : 3 vertex nodes in 2D, or all edge nodes in 3D, middle nodes m j : all mid-edge nodes in 2D, or mid-triangle nodes in 3D, internal nodes i j : all internal nodes in the triangle/tetrahedra. The three types of nodes are illustrated in Figs. 1 and 2. In simple words, {c j } are nodes shared by possibly more than two elements, {m j } are nodes shared by no more than two elements, and {i j } are nodes internal to one element. A Lagrange nodal basis function φ j is a P k+2 polynomial which assumes value 1 at one node c j, but vanishes at all other dim Pk+2 d 1 nodes. For example, a P2 4 nodal basis function on the reference triangle {0 x, y, 1 x y 1}, at node (1/4, 0), c.f. Fig. 3, is x(1 x y)(3/4 x y)(2/4 x y) φ 2 (x, y) = (1/4)(1 1/4 0)(3/4 1/4 0)(2/4 1/4 0). (46) The restriction of a nodal basis φ j on a lower dimensional simplex, a triangle or an edge or a vertex, is also a nodal basis function on that lower dimensional finite element. For example, this node basis function (46) is the restriction of the following 3D nodal basis function (at node (1/4, 0, 0) on tetrahedron {0 x, y, z, 1 x y z 1}) on the reference triangle, x(1 x y z)(3/4 x y z)(2/4 x y z) φ j (x, y, z) = (1/4)(1 1/4 0 0)(3/4 1/4 0 0)(2/4 1/4 0 0). (47) The restriction of 2D basis function φ 2 in (46) in 1D is, c.f. Fig. 3, φ j (x) = x(1 x)(3/4 x)(2/4 x) (1/4)(1 1/4)(3/4 1/4)(2/4 1/4). (48)

17 J Sci Comput (2014) 59: Fig. 3 A 2D nodal basis φ 2 (46) and its restriction in 1D, φ j (48) Fig. 4 A 1D nodal basis φ j (48) and its dual basis in 1D, ψ [1D] j (50), 1 0 φ j ψ [1D] j = 1 On each element T (an edge, a triangle, or a tetrahedron), the P k Lagrange basis {φ j } has a dual basis {ψ j Pk d }, satisfying { 1 if j = j φ j ψ j dx = δ jj =, 0 if j = j (49). T In other words, if writing {ψ j } as linear combinations of Lagrange basis {φ j }, the coefficients are simply the inverse matrix of the mass matrix, the L 2 -matrix of {φ j }. For example, the dual basis function ψ 2 for the nodal basis function φ 2 in (46)(2D)is ψ [2D] 2 (x, y) = x x x x 2 y xy2 1890x x 3 y 5670x 2 y xy 3. We can compute the dual of ψ j in (48) in1dtoget ψ [1D] j (x) = x x x x4 (50) = φ φ φ φ φ 4, where φ i is the nodal basis on [0, 1] at x i = i/4, i = 0, 1, 2, 3, 4. The dual function ψ [1D] j (x) in (50) is plotted in Fig. 4. Similarly we can compute the dual basis function for (47)in3D. We note that both Lagrange nodal basis and its dual basis are affine invariant. That is, the Lagrange basis on the reference triangle is also the Lagrange basis on a general triangle after

18 490 J Sci Comput (2014) 59: an affine mapping. For simplicity, we use the same notations φ j and ψ [2D] j for the nodal basis and the dual basis functions on the reference triangle and on a general triangle. We now define the Scott-Zhang interpolation operator: Q 0 : H 1 ( ) V h, where V h is the C 0 -P k+2 finite element space defined in (9). Q 0 v is defined by the nodal values at three types nodes. 1. For each corner node c j (shared by possibly more than two elements), we select one boundary (d 1)-dimensional simplex C j if c j is a boundary node, or any one (d 1)- dimensional face simplex C j on which the node is, as c j s averaging patch. C.f. Fig. 1, the boundary node c j has a boundary edge C j, while the corner node c j can choose any one of four edges passing it, as its averaging patch. In Fig. 2, a corner node c j has a triangle C j as its averaging patch. In both 2D and 3D, we use a same definition Q 0 v(c j ) = C j [(d 1)D] ψ j v(x)dx. (51) 2. For each middle node m j, the averaging patch C j is the unique (d 1)-dimensional simplex C j containing m j,seefigs.1and 2. The interpolated nodal value is then determined by the unique solution of linear equations: ( ) Q 0 v(m j )φ j (x) p i (x)dx (52) m j C j C j = C j ( v(x) ) Q 0 v(c j )φ j (x) c j C j p i (x)dx for all degree (k + 2 d) polynomials p i on (d 1)-simplex C j,whereq 0 v(c j ) is defined in (51). In 2D, c.f. Fig. 1, after we determine the nodal values at the two end points (c j ), we determine the middle-edge points nodal value by (52). 3. After determine all nodal values on the surface of each element, we define the interpolation inside the element: The nodal values at internal nodes (Q 0 (i j )) are determined by the unique solution of the following linear equations ( ) p i dx (53) C j Q 0 v(i j )φ j i j C j ( = v Q 0 v(m j )φ j Q 0 v(c j )φ j m j C j c j C j C j for all degree (k + 1 d) polynomials p i on d-simplex C j. By (51) (53), the (refined) Scott-Zhang interpolation is Q 0 v = ) p i dx x j N h Q 0 v(x j )φ j (x), (54) where N h is the set of all C 0 -P k+2 Lagrange nodes of triangulation T h.

19 J Sci Comput (2014) 59: Fig. 5 Lagrange nodal basis φ s j on subinterval (d = 2) or subtriangle (d = 3), c.f., (57) Remark 8.1 If all corner nodes on a (d 1)-simplex C j have selected C j as their averaging patch in (51), then the solution of (52) isthel 2 -projection of v on C j, i.e., [(d 1)D] Q 0 v(m j ) = ψ j v(x)dx. (55) C j Lemma 8.1 The Scott-Zhang interpolation operator (54) is well-defined, i.e., the linear systems of Eqs. (52) and (53) both have unique solutions. Proof For the linear system of Eq. (52), we change the Pk+2 d d 1 basis functions ({p i = 1, x,..., x k } when d = 2, or {p i = 1, x, y, x 2, xy,..., y k 1 } when d = 3) uniquely as linear combinations of the Lagrange basis functions on the subinterval C s j (d = 2) or the subtriangle C s j (d = 3), c.f. Fig. 5. p i = ( ) c i, j φ s j, i = 1, 2,..., dim Pk+2 d d 1. (56) j The nodal basis functions on a simplex C j and its subsimplex C s j differ by a bubble functions: φ j = φ s b(x) j b(x j ), (57) where b(x) is the bubble functions assuming 0 on the boundary of C j. For example, c.f. Fig. 5,whend = 2andC j =[0, 1], b(x) = x(1 x), φ1 s (x 2/4)(x 3/4) (x) = (1/4 2/4)(1/4 3/4), φ 2 (x) = φ1 s b(x) (x) b(1/4). By the change of basis, (56)and(57), the linear system (52) is equivalent to the following weighted-mass linear systems: Q 0 v(m j ) φ j φi s dx = vφi s dx Q 0 v(c j ) φ j φi s dx, (58) m j C j C j C j j C j C j i = 1, 2,..., dim ( ) Pk+2 d d 1.

20 492 J Sci Comput (2014) 59: The coefficient matrix in (58) is the mass matrix on simplex C j (c.f. Fig. 5) with a positive weight: a i, j = φi s φ j dx = φi s φs j w(x)dx, C j C j where w(x) = b(x) b(x j ) > 0 in interior(c j ). As the Lagrange basis {φi s } (on the subsimplex) are linearly independent, the mass (with weight) matrix in (58) is invertible, and the equivalent linear system (52) has a unique solution too. For example, when d = 2andk = 2, the coefficient matrix in (58) and its inverse are 152/315 16/63 8/63 4/21 18/35 4/21 8/63 16/63 152/ / /64 225/1024 = 225/256 45/16 225/ / / /1024 By the same argument, lifting the space dimension by 1, we can show that (53) has a unique solution too. In fact, the system (53) whend = 2 is the same system (52) with d = 3there. Lemma 8.2 If d = 2, the Scott-Zhang interpolation operator (54) preserves the volume mass of order k 1 and the face mass of order k, i.e., (v Q 0 v)p i dx = 0 T T h, p i P k 1 (T ), (59) T (v Q 0 v)p i dx = 0 E E h, p i P k (E), (60) E where E h is the set of edges in triangulation T h. Proof By the construction (52), we have ( (v Q 0 v)p i dx = v Q 0 v(m j )φ j Q 0 v(c j )φ j )p i dx = 0. E E m j E That is (60). Here, because we have two missing dof (degrees of freedom) at the two end points of each edge, the polynomial degree in mass preservation is reduced by two, from (k + 2) to k. Similarly, (59) follows (53). Here, the polynomial degree reduction is three as each triangle has three edges where the interpolation values are not free (not determined by (53)). Lemma 8.3 If d = 3, the Scott-Zhang interpolation operator (54) preserves the volume mass of order k 2 and the face mass of order k 1, i.e., (v Q 0 v)p i dx = 0 T T h, p i P k 2 (T ), (61) T c j E (v Q 0 v)p i dx = 0 E E h, p i P k 1 (E), (62) E where E h is the set of face triangles in the tetrahedral grid T h.

21 J Sci Comput (2014) 59: Proof As each triangle E has three edges, where the interpolation is not determined possibly by function value on neighboring triangles, we lose dof s on the three edges in the interpolation. That is, we lose three orders in face-mass conservation in (52). (62) issim- ply another expression of (52), as in the proof of Lemma 8.2. By(53), (61) follows. Here the polynomial-degree deduction in mass conservation is 4, due to 4 face-triangles each tetrahedron. Remark 8.2 By Lemmas 8.2 and 8.3, the mass preservation on element and on faces is one order higher than the requirement (16), when d = 2. When d = 3, (61)and(62)imply(16). Theorem 8.1 The Scott-Zhang interpolation operator (54) is of optimal order in approximation, i.e., when k 1, v Q 0 v +h v Q 0 v 1 Ch k+3 v k+3 v H k+3 ( ). (63) Further, when k 0, 1/2 h 2 v Q 0 v 2 H 2 (T ) Ch k+3 v k+3 v H k+3 ( ). (64) Proof The Scott-Zhang operator preserves a degree (k + 2) polynomial on the star union of an element T, S T = T T = T, T, T T h. That is, Q 0 v = v v Pk+2 d (S T ), (65) when k 1. (65) is shown in three steps. First, by (51), when v P k+2, the dual basis defines Q 0 v(c j ) = v(c j ), (66) at all corner nodes. In the second step, by (66) and(58), (52) holds for p i = ψi s too where m i are all middle nodes on C j.thatis, C j By (66), as v P k+2, ( v Q 0 (c j )φ j Q 0 v(m j )φ j j C j m j ) φi s dx = 0. (67) v j C j Q 0 (c j )φ j = v c b(x) for some v c P d 1 k+2 d, where b(x) is a bubble function, cf. (57). For the middle node basis functions, we have also, c.f. (57), φ j = φ s j b(x)/b(x j ) m j C j. Thus (67) implies, where m Q j 0 v(m j )φ j = v m b(x) for some v m Pk+2 d d 1, (v b v m )φi s b(x)dx = 0, C j v b v m 0 and v = Q 0 v on C j.

22 494 J Sci Comput (2014) 59: Therefore, at all middle nodes, Q 0 v(m j ) = v(m j ). (68) In the third step, by (53) and the same argument in the second step, Q 0 (i j ) = v(i j ), at all internal nodes. Thus Q 0 v = v P k+2. We then use the standard scaling argument (on the dual basis functions) and the Sobolev inequality, as in Theorem 3.1 of [26], it follows that Q 0 v H 1 (T ) C v H 1 (S T ) v H 1 ( ). The above stability result leads directly to the optimal-order approximation (63), following the standard argument (i.e., by (65) and the existence of local Taylor polynomials, c.f. for example, [6]), as shown in Theorem 4.1 of [26]. We note again that the Scott-Zhang operator here is a refined version of the Scott-Zhang operator in [26]. After showing the local preservation of P k+2 polynomials above, the proof of the theorem is the same as that in [26]. (64) is (4.4) in [26], with p = q = 2, m = 2, and l = k + 3there. References 1. Adolfsson, V., Pipher, J.: The inhomogeneous Dirichlet problem for 2 in Lipschitz domains. J. Funct. Anal. 159(1), (1998) 2. Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, (1968) 3. Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Model. Math. Anal. Numer. 19(1), 7 32 (1985) 4. Bell, K.: A refined triangular plate bending element. Int. J. Numer. Methods Eng. 1, (1969) 5. Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2(4), (1980) 6. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, third edition. Texts in Applied Mathematics, 15. Springer, New York (2008) 7. Brenner, S., Sung, L.: C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, (2005) 8. Clough, R.W., Tocher, J.L.: Finite element stiffness matrices for analysis of plates in bending. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics. Wright Patterson A.F.B, Ohio (1965) 9. Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics, Springer, Berlin (1988) 10. Douglas Jr, J., Dupont, T., Percell, P., Scott, R.: A family of C 1 finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO Anal. Numer. 13(3), (1979) 11. Engel, G., Garikipati, K., Hughes, T., Larson, M.G., Mazzei, L., Taylor, R.: Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Meth. Appl. Mech. Eng. 191, (2002) 12. Falk, R.: Approximation of the biharmonic equation by a mixed finite element method. SIAM J. Numer. Anal. 15, (1978). Numer. Anal. 15, pp (1978) 13. Fraeijs de Veubeke, B.: A conforming finite element for plate bending. In: Zienkiewicz, O.C., Holister, G.S. (eds.) Stress Analysis, pp Wiley, New York (1965) 14. Girault, V., Raviart, P.A.: Finite Element Methods for the Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin (1986) 15. Gudi, T., Nataraj, N., Pani, A.K.: Mixed discontinuous Galerkin finite element method for the Biharmonic equation. J. Sci. Comput. 37(2), (2008)

23 J Sci Comput (2014) 59: Hu, J., Huang, Y., Zhang, S.: The lowest order differentiable finite element on rectangular grids. SIAM Numer. Anal. 49(4), (2011) 17. Hu, J., Shi, Z.-C.: The best L2 norm error estimate of lower order finite element methods for the fourth order problem. J. Comput. Math. 30(5), (2012) 18. Monk, P.: A mixed finite element methods for the biharmonic equation. SIAM J. Numer. Anal. 24, (1987) 19. Morley, L.S.D.: The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19, (1968) 20. Mozolevski, I., Süli, E.: hp-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation. J. Sci. Comput. 30(3), (2007) 21. Mu, L., Wang, J., Wang, Y., Ye, X.: Weak Galerkin mixed finite element method for the biharmonic equation, arxiv: Mu, L., Wang, J., Ye, X.: A computational study of the weak Galerkin method for the second order elliptic equations. Numer. Algorithm doi: /s Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, preprint 24. Percell, P.: On cubic and quartic Clough-Tocher finite elements. SIAM J. Numer. Anal. 13, (1976) 25. Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3 4, (1977) 26. Scott, L., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, (1990) 27. Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems, arxiv: v1 28. Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, (2013) 29. Ženišek, A.: Alexander polynomial approximation on tetrahedrons in the finite element method. J. Approx. Theory 7, (1973) 30. Zhang, S.: A C1 P2 finite element without nodal basis. M2AN 42, (2008) 31. Zhang, S.: A family of 3D continuously differentiable finite elements on tetrahedral grids. Appl. Numer. Math. 59(1), (2009) 32. Zhang, S.: On the full C 1 Q k finite element spaces on rectangles and cuboids. Adv. Appl. Math. Mech. 2, (2010) 33. Zlámal, M.: On the finite elmeent method. Numer. Math. 12, (1968)