A New Class of Zienkiewicz-Type Nonconforming Element in Any Dimensions

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1 Numerisce Matematik manuscript No. will be inserted by te editor A New Class of Zienkiewicz-Type Nonconforming Element in Any Dimensions Wang Ming 1, Zong-ci Si 2, Jincao Xu1,3 1 LMAM, Scool of Matematical Sciences, Peking University, mwang@mat.pku.edu.cn 2 Institute of Computational Matematics, CAS, si@lsec.cc.ac.cn 3 Department of Matematics, Pennsylvania State University, xu@mat.psu.edu, ttp:// Te date of receipt and acceptance will be inserted by te editor Summary In tis paper, a new class of Zienkiewicz-type nonconforming finite element, in n spatial dimensions wit n 2, is proposed. Te new finite element is proved to be convergent for te biarmonic equation. Key words Nonconforming finite element, fort order elliptic equation, biarmonic 1 Introduction In tis paper, we will propose a new class of Zienkiewicz-type nonconforming simplex finite element for n-dimensional fourt order partial differential equations wit n 2. It uses te values of function and first order derivatives at vertices as degrees of freedom, tat is, it uses te same degrees of freedom wit te Zienkiewicz element [2 or 6]. But its sape function space is different from te one of te Zienkiewicz element. As a nonconforming finite element for fourt order partial differential equations, te Zienkiewicz element is attractive. Te first ting is its convergent property. In two dimensional case, te Zienkiewicz element is only convergent under te parallel line condition, and is divergent in general grids. Te numerical experiments were given in [7] and te matematical proof can be found in [10]. Anoter attractive ting is te degrees of freedom of te Zienkiewicz element. It is convenient for numerical computations to take values of function and derivatives at vertices as degrees of freedom. Altoug on eac single element te number of degrees of freedom of te Zienkiewicz element is not te least, te global number is. Te work was supported by te National Natural Science oundation of Cina Tis work was supported in part by NS DMS and NS DMS and te Cangjiang Professorsip troug Peking University

2 2 Wang M, Z-C Si, J Xu Tere are some modified triangular elements, wit te same degrees of freedom of te Zienkiewicz element, proposed by different ways, suc as, te TQC9 element by te quasiconforming metod [18,5], te generalized conforming element by te generalized conforming metod [9], te TRUNC element by te free formula metod [1,4] and te Bergan element by te energy ortogonal metod [3]. We call tese elements are of Zienkiewicztype, Z-type for sort. Teir convergence analysis were given in [20,13,11,14] respectively. It is a little surprise tat tere were no convergent Z-type element proposed directly from te nonconforming element metod in two dimensions. In tree dimensional case, two convergent Z-type elements were proposed in [19]. One is constructed by te quasi-conforming metod, and anoter is a non C 0 nonconforming element wic is reduced from a cubic tetraedral nonconforming element proposed also in [19]. or tis cubic element, te number of te degrees of freedom is just te dimension of te cubic polynomial space. It does not occur in oter dimensional cases wen te similar degrees of freedom are used. Te new Z-type element proposed in tis paper is a nonconforming C 0 element, and it is constructed in a canonical fasion for two and iger dimensions. Te rest of te paper is organized as follows. Section 2 recalls te nonconforming element metod. Section 3 gives a detailed descriptions of a new Z-type nonconforming element. Section 4 sows te convergence of te new element. Te last section gives some concluding remarks. 2 Preliminaries Let Ω be a bounded polyedroid domain in R n n 2 wit boundary Ω. or a nonnegative integer s, Let H s Ω, H0 sω, s,ω and s,ω denote te usual Sobolev spaces, norm and semi-norm respectively. Let, denote te inner product of L 2 Ω. or f L 2 Ω, we consider te following fourt order boundary value problem: 2 u = f, in Ω, u Ω = u 1 =0 ν Ω were ν =ν 1,ν 2,,ν n is te unit outer normal to Ω and is te standard Laplacian operator. Set Define av, w = = n Ω i,j=1 x 1, 2 v x i x j x 2,,. x n 2 w x i x j, v, w H 2 Ω. 2 Te weak form of problem 1 is: find u H0 2 Ω suc tat au, v =f,v, v H0 2 Ω. 3

3 Zienkiewicz-Type Nonconforming Element 3 or a subset B R n and a nonnegative integer r, let P r B be te space of all polynomials wit degree not greater tan r. Let T,P T,Φ T be a finite element were T is te geometric sape, P T te sape function space and Φ T te vector of degrees of freedom, and let Φ T be P T -unisolvent see [6]. Let T be a triangulation of Ω wit mes size. or eac element T T, let T be te diameter of te smallest ball containing T and ρ T be te diameter of te largest ball contained in T. Let {T} be a family of triangulations wit 0. Trougout te paper, we assume tat {T} is quasi-uniform, namely it satisfied tat T ηρ T, T T for a positive constant η independent of. or eac T, let V and V 0 be te corresponding finite element spaces associated wit T,P T,Φ T for te discretization of H 2 Ω and H0 2 Ω respectively. In te case of nonconforming element, V H 2 Ω and V 0 H0 2Ω. or v, w L 2 Ω tat v T,w T H 2 T, T T, we define a v, w = n T i,j=1 2 v x i x j 2 w x i x j. 4 Te finite element metod for problem 3 corresponding to te element T,P T,Φ T is: find u V 0 suc tat a u,v =f,v, v V 0. 5 or any v L 2 Ω tat v T H m Ω, T T, we define te following mesdependent norm m, and semi-norm m, : 1/2, 1/2. v m, = v 2 m,t v m, = v 2 m,t or nonconforming elements, te basic matematical teory as been establised see [6,8,15-17]. We will use tem to discuss te convergence of te new element. 3 A New Z-Type Nonconforming Element In tis section, we give a detailed description of our new Z-type nonconforming element in n-dimensions n 2. Given an n-simplex T wit vertices a i, 1 i n +1, denote by i 1 i n +1 te n 1-dimensional subsimplex of T witout a i as its vertex, and by λ 1,λ 2,,λ n+1 te barycentric coordinates of T. Denote by T and i te measures of T and i respectively. Define P 3T =P 2 T +span{λ 2 i λ j λ i λ 2 j 1 i<j n +1}. Ten te sape function space of te n-dimensional Zienkiewicz element is just P 3 T.

4 4 Wang M, Z-C Si, J Xu Now let q 0 be te bubble function defined by or 1 i<j n +1, we define q ij = λ 2 i λ j λ i λ 2 j + 2n 1! n! q 0 = λ 1 λ 2 λ n+1. n 1n n+1 λ i λ j + 1 k n+1 k i,k j λ i λ j λ k nλ λ k 2 k 1 q 0. 6 A new Z-type nonconforming element, NZT element for sort, is defined by T,P T,Φ T wit 1 T is an n-simplex. 2 P T = P 2 T +span{ q ij 1 i<j n +1}. 3 Te components of Φ T are: va j, 1 j n +1, a j a i va i, 1 i j n +1, v C 1 T. Te degrees of freedom of te NZT element are just te same wit te n-dimensional Zienkiewicz element see ig. 1. a 3 a 3 a 1 a 2 a 1 a 2 n =2 n =3 ig. 1 Let ν i denote te unit out normal of n 1-subsimplex i of T 1 i n +1. By certain computation, we can obtain tat 1 p i ν i = 1 p n ν i a k, 1 i n +1, p P T. 7 i 1 k n+1 k i Lemma 1 or NZT element, Φ T is P T -unisolvent. Proof Let p P T and pa j =0, 1 j n +1; a j a i pa i =0, 1 i j n +1. We only need to sow tat p 0. Let q 1,,q l be a basis of P 2 T. Ten p can be wriiten as l p = c i q i + c ij q ij i=1 1 i<j n+1 a 4

5 Zienkiewicz-Type Nonconforming Element 5 were c i and c ij are constants. Set q ij = λ 2 i λ j λ i λ 2 j and l p = c i q i + c ij q ij. i=1 1 i<j n+1 Ten It can be verified tat p = p + c ij q ij q ij. 1 i<j n+1 q ij q ij a k =0, 1 k n+1; a m a k q ij q ij a k =0, 1 k m n+1. Tus, p satisfies pa j =0, 1 j n +1; a j a i pa i =0, 1 i j n +1. On te oter and, p P 3 T. Tus p 0, tat is, c i =0, 1 i l; c ij =0, 1 i<j n +1. It follows tat p 0. or 1 i j n +1we define p ij = 1 2 λ iλ j 1 + λ i λ j, + 2n 1! 2n! p i = λ 2 i +2 n 1n n+1 λ i λ j + 1 j n+1 j i p ij. 1 k n+1 k i,k j λ i λ j λ k nλ λ k 2 k 1 q 0 8 Let δ ij be te Kronecker delta. It can be verified tat for 1 i j n +1and 1 k l n +1, { pi a k =δ ik, a l a k p i a k =0, 9 p ij a k =0, a l a k p ij a k =δ ik δ jl. Tat is, p i and p ij are te nodal basis functions respect to te degrees of freedom. Te corresponding interpolation operator Π T can be written by, Π T v = p i va i + p ij a j a i va i, v C 1 T i n+1 1 i j n+1 or NZT element, we can define te corresponding finite element spaces V and V 0 as follows: V = {v L 2 Ω v T P T, T T, v and v are continuous at all vertices of elements in T}. V 0 = {v V v and v vanis at all vertices belonging to Ω}.

6 6 Wang M, Z-C Si, J Xu We claim tat V H 1 Ω and V 0 H0 1Ω. Let v V, and let be a common n 1-dimensional subsimplex of T,T T. By definition, te restrictions of v T and v T on are all in P 3, and tey and teir first order derivatives are equal at all vertices of respectively. Tus v T = v T on, tat is, v C 0 Ω, and tis leads to tat v H 1 Ω. Using similar argument, we can sow v H0 1Ω wen v V 0. Given any n 1-dimensional subsimplex and v V, let us define te jump of v across as follows: [ v ]= v T v T if = T T for some T,T T and [ v ]= v T if = T Ω. Te following lemma is a direct consequence of equality 7 and te definitions of V and V 0. Lemma 2 Let V and V 0 be te finite element spaces corresponding to NZT element. If is a common n 1-dimensional subsimplex of T,T T, ten [ v ]=0, v V. 11 If an n 1-dimensional subsimplex of T T is on Ω ten [ v ]=0, v V Remark. Let V and V 0 be te finite element spaces corresponding to NZT element. By Lemma 2 and Green s formula, we can obtain directly tat a p, v =0, p P 2 Ω, v V 0. We know from [15] tat te NZT element passes te patc test on triangulationt. 4 Convergence Analysis In tis section, we discuss te convergence property of NZT element. Let V and V 0 be te finite element spaces corresponding to NZT element. irst, we consider te error estimates for finite element spaces. Teorem 1 Let V and V 0 be te finite element spaces corresponding to NZT element. Ten tere exists a constant C independent of suc tat inf v V 0 m=0 3 m v v m, C 3 v 3,Ω, v H 3 Ω H0Ω, 2 13 inf v V m=0 3 m v v m, C 3 v 3,Ω, v H 3 Ω. 14

7 Zienkiewicz-Type Nonconforming Element 7 Proof or v H 3 Ω H0 2Ω, let w L 2 Ω suc tat T T, w T P 2 T and qw dx = qvdx, q P 2 T. By te interpolation teory, we ave T T v w m, C 3 m v 3,Ω, 0 m Given a set B R n, let TB ={ T T B T }and N B be te number of te elements in TB. Now we define v V 0 as follows: for any T T, if vertex a i of T is in Ω ten v a i = 1 N a i T Ta i w T a i, v a i = Obviously, v is well-defined. We will sow tat 1 N a i T Ta i w T a i. v v m, C 3 m v 3,Ω, 0 m Let T T, by a standard scaling argument, we ave n+1 p 2 m,t Cn 2m pa i pa i 2, 0 m 3, p P T. 17 i=1 Set φ = w v. Obviously, φ T P T. If vertex a i of T is in Ω, te definition of v leads to tat 1 φ T a i = w T a i w T a i. N a i T Ta i or T Ta i tere exist T 1,,T J Ta i suc tat T 1 = T, T J = T and j = T j T j+1 is a common n 1-dimensional subsimplex of T j and T j+1 and a i j, 1 j<j. By te inverse inequality, we ave w T a i w T a i 2 J 1 = w Tj a i w Tj+1 a i 2 j=1 J 1 C w Tj a i w Tj+1 a i j=1 J 1 C 1 n 2 w Tj w Tj+1 j=1 J 1 C 1 n v 2 w Tj j=1 0, j + 0, j 2 2 v w Tj+1 0, j

8 8 Wang M, Z-C Si, J Xu By te interpolation teory, we obtain Since N T is bounded, we ave w T a i w T a i 2 C 6 n φ T a i 2 C 6 n T TT J j=1 v 2 3,T j v 2 3,T 18 If vertex a i of T is on Ω ten tere exists T Ta i wit an n 1-dimensional subsimplex of T on Ω and a i. By te definitions of w and v,weave φ T a i = w T a i w T a i +w T a i w T a i w T a i + w T a i. By te inverse inequality and te interpolation teory w T a i 2 C 1 n w T 2 0, = C 1 n v w T 2 0, C 6 n v 2 3,T. By a similar analysis for w T a i w T a i, we conclude tat 18 is also true in tis case. Similarly, we can sow tat n+1 φ T a i 2 C 4 n i=1 Combining 17, 18 and 19, we ave 2m φ 2 m,t C6 T TT T TT v 2 3,T. Summing te above inequality over all T T, we obtain tat 2m φ 2 m, C6 v 2 3,T. T TT v 2 3,T. 19 Consequently 2m φ 2 m, C6 v 2 3,Ω. 20 Inequality 16 follows from 20 and 15. Using similar argument, we can sow 14. Lemma 3 Let V 0 be te finite element space corresponding to NZT element. Ten tere exists a constant C independent of suc tat for v H 3 Ω a v, v 2 v, v C v 3,Ω v 2,, v V 0. 21

9 Zienkiewicz-Type Nonconforming Element 9 Proof Let v V 0 and φ H 1 Ω.GivenT T and an n 1-dimensional subsimplex of T, let P 0 : L2 P 0 be te L 2 -ortogonal projection. Let i, j {1, 2,, n}. By Lemma 2 and Green s formula we ave T φ 2 v + φ v = x i x j x i x j = T T = T φ v x j ν i = T v φ P 0 v ν i x j x j φ v x j ν i φ P 0 φ v P 0 v ν i x j x j Using te Scwarz inequality and te interpolation teory we obtain tat φ P 0 v φ P 0 v ν i x j x j T T C φ P 0 φ 0, v P 0 x j φ 1,T v 2,T C φ 1,Ω v 2,. v x j 0, Consequently, for i, j {1, 2,,n}, φ 2 v + φ v C φ 1,Ω v 2,, φ H 1 Ω, v V x i x j x i x j T Using 22 and te following equality, n a v, v 2 v, v = v 2 v i=1 T T x 2 + v v i x i x i T + 2 v 2 v + 3 v 1 i j n T T x i x j x i x j x 2 i x j T 2 v 2 v T x 2 i x 2 j 1 i j n we obtain te conclusion of te lemma. + 3 v v x 2 i x, j x j v x j Teorem 2 Let V 0 be te finite element space corresponding to NZT element, and let u and u be te solutions of problems 3 and 5 respectively. Ten 23 lim u u 2, =0, 24 0

10 10 Wang M, Z-C Si, J Xu and tere exists a constant C independent of suc tat wen u H 3 Ω. u u 2, C u 3,Ω 25 Proof rom Lemma 2 we see tat NZT element passes te -E-M-Test in [12]. Hence NZT element passes te generalized patc test. By Teorem 1 and te fact tat H0 2 Ω is te closure of C0 Ω in norm 2,Ω, we obtain lim inf v v 2, =0, v H0Ω. 2 0 v V 0 Tus 24 is true by te result in [16]. By te generalized Poincare-riedrics inequality [17] and te Strang Lemma see [6] or [15], we ave a u, w f,w u u 2, C inf u w 2, + sup. w V 0 w 2, Ten 25 follows from 13 and 21. w V 0 w 0 5 Concluding remarks To construct a convergent Z-type nonconforming element for te fourt order elliptic boundary value problems, is motivated by te teoretical interest and te efficiency consideration in practical computation. In tis paper, te NZT element, a new n-dimensional C 0 nonconforming simplex finite element, is constructed and analyzed. Te NZT element uses te same degrees of freedom wit te Zienkiewicz element and te different sape function space. Unlike te Zienkiewicz element, te NZT element is convergent and passes te patc test in general grids. References 1. Argyris J H, Haase M and Mlejnek H P, On an unconventional but natural formation of a stiffness matrix, Comput. Mets. Appl. Mec. Eng., , Bazeley G P, Ceung Y K, Irons B M and Zienkiewicz O C, Triangular elements in plate bending conforming and nonconforming solutions, in Proceedings of te Conference on Matrix Metods in Structural Mecanics, Wrigt Patterson A.. Base, Oio, 1965, Bergan P G, inite elements based on energy ortogonal functions, Int. J. Numer. Mets. Eng., , Bergan P G and Nygard M K, inite elements wit increased freedom in coosing sape functions, Int. J. Numer. Mets. Eng., , Cen Wanji, Liu Yingxi and Tang Limin, Te formulation of quasi-conforming elements, J. Dalian Inst. of Tecnology, 19, , Ciarlet P G, Te inite Element Metod for Elliptic Problems, Nort-Holland, Amsterdam, New York, 1978.

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