Numerische Mathematik

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1 Numer. Math. (00) 98: Digital Object Identifier (DOI) /s Numerische Mathematik On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation Shangyou Zhang Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA; Received March 7, 003 / Revised version received January 9, 00 / Published online July 1, 00 c Springer-Verlag 00 Summary. It is shown in this paper that the optimal approximation property of bilinear or trilinear finite elements would be retained when the affine mapping is used to replace the Q 1 mapping on each element, if the grids are refined nestedly. The new method truncates the quadratic and cubic terms in reference mappings and produces constant Jacobians and Jacobian matrices. This would avoid a shortcoming of the quadrilateral and hexahedral elements where the integrals of rational functions have to be computed or approximated. Numerical tests verify the analysis. Mathematics Subject Classification (000): 65N30, 65N50, 65N55 1 Introduction Comparing to the triangular and tetrahedral finite elements, quadrilateral and hexahedral elements have certain advantages. For example, we have much less elements and we could evaluate integrals and solve the resulting linear systems quicker. However, one disadvantage is that instead of integrals of polynomials on the reference triangular or tetrahedral element, we may have to evaluate integrals of rational functions. This is because, for nonparallelogram or non-parallelepiped elements, the reference mappings are Q 1 functions and the resulting Jacobian determinants and inverse Jacobian matrices would be no longer constant. In fact, they are rational functions. Unlike the case of integration of polynomials, which can be done easily by product formulas of Gauss quadratures on the reference square or cube, the numerical integration of rational functions is relatively difficult. Therefore,

2 560 S. Zhang some exact formulas were provided for certain rational functions on squares (cf. [11 1]) for D elements only. They do provide practical help, but they are far from completeness. Recently, Arunakirinathar and Reddy proposed to replace Q 1 reference mappings by affine mappings in D (cf. [3 5].) Under the condition that each quadrilateral would approach some parallelogram at a higher order of its size h, Arunakirinathar and Reddy proved the optimal convergence rate of finite element solutions after replacing the Q 1 reference mappings by affine mappings in [5]. In this paper, we restrict the method to a typical situation that the grids are refined nestedly in D and 3D. We will show that under the standard nested refinement, both sub-quadrilaterals and sub-hexahedra will converge to parallelograms and parallelepipeds at the second order of the element size (see Definition.1 and Theorem.1 below.) We then will show that, both in D and 3D, we can replace Q 1 reference mappings by affine mappings while retaining the optimal order of convergence. Our theory truly extends that of [5] to 3D. The condition assumed in [5], i.e., quadrilateral elements converge to parallelograms at a higher order of the element size, would guarantee (0) () in 3D, while we will prove them under nested refinement condition, not only in D but also in 3D. The D nested refinement result, Theorem.1 below, was discovered by Zhang [19] first, according to the author s knowledge, and it was stated in [0]. In [10] and references therein, Ming and Shi studied several definitions of shape regularity of quadrilateral meshes and showed their equivalence. Under such shape regularity conditions, i.e., asymptotically parallelogram meshes, Wilson s elements [15], serendipity elements [] and nonconforming rotated Q 1 elements [10] were shown to converge at their optimal order. This paper extends the above D nested refinement to 3D. Some results above are ready to be generalized to 3D. Even superconvergence may be studied under the nested refinement. Under some strong conditions (shape regularity and local symmetry) on quadrilateral meshes, Zlamal showed some superconvergence convergence results in [1]. More general theories of superconvergence on locally uniform and other types of meshes, both in D and higher dimensions, are summarized in [16]. Nested refinement In this section, we consider the nested refinement of general quadrilaterals and general hexahedra (see Figure 1), where we use middle (center) points to subdivide each element into half-sized elements, as shown in Figures and 3. We note that the quadrilateral and hexahedral elements in this paper are defined via the reference elements and the Q 1 reference mappings. Therefore, the faces of hexahedral elements are not necessarily flat, see the top

3 On the nested refinement of quadrilateral and hexahedral finite elements 561 v 1 v v 3 s(k) v Not planar. Fig. 1. A general quadrilateral and a hexahedron Fig.. The standard nested refinement of a quadrilateral face in Figure 3 for an example. So we refer the quadrilateral and hexahedral elements as -vertex and 8-vertex elements sometimes, to be more accurate. We use the reference elements in D and 3D, as shown in Fig. : ˆK = [ 1, 1] n, n =, 3. The reference mapping from the reference element to a general element is defined by the Q 1 nodal basis functions and the coordinates of vertices. For n =, the Q 1 nodal basis functions on the reference element are (see Fig. for the vertex numbering scheme) b 1 ( ˆx,ŷ) = (1 ˆx)(1 ŷ)/, b ( ˆx,ŷ) = (1 +ˆx)(1 ŷ)/, (1) b 3 ( ˆx,ŷ) = (1 +ˆx)(1 +ŷ)/, b ( ˆx,ŷ) = (1 ˆx)(1 +ŷ)/. Given vertices, v i = (x i,y i ), i = 1,, 3,, of a strictly convex quadrilateral K in D, we can define a mapping: F : ˆK K, F : ( ˆx,ŷ) (x, y),

4 56 S. Zhang Fig. 3. The standard nested refinement of a hexahedron where () ( x = y) v i b i ( ˆx,ŷ) = a 00 + a 10 ˆx + a 01 ŷ + a 11 ˆxŷ. i=1 The vector coefficients are (3) a 00 = v 1 + v 3 + v + v a 01 = v v 1 + v 3 v,a 10 = v v 1 + v 3 v,, a 11 = v 1 + v 3 v v. Similarly, the elements in 3D are defined. The Q 1 nodal basis functions in 3D are (see Figure for the vertex numbering method) () b 1 = (1 ˆx)(1 ŷ)(1 ẑ)/8, b = (1 +ˆx)(1 ŷ)(1 ẑ)/8, b 3 = (1 +ˆx)(1 +ŷ)(1 ẑ)/8, b = (1 ˆx)(1 +ŷ)(1 ẑ)/8, b 5 = (1 ˆx)(1 ŷ)(1 +ẑ)/8, b 6 = (1 +ˆx)(1 ŷ)(1 +ẑ)/8, b 7 = (1 +ˆx)(1 +ŷ)(1 +ẑ)/8, b 8 = (1 ˆx)(1 +ŷ)(1 +ẑ)/8. Given 8 vertices, v i = (x i,y i,z i ), i = 1,,...,8, in 3D, we define an 8-vertex element by the following 3D Q 1 mapping: F : ˆK K := F( ˆK), F : ( ˆx,ŷ,ẑ) (x,y,z), where x y = z 8 v i b i ( ˆx,ŷ,ẑ) i=1 = a a 100 ˆx + a 010 ŷ + a 001 ẑ + a 110 ˆxŷ + a 101 ˆxẑ + a 011 ŷẑ (5) +a 111 ˆxŷẑ.

5 On the nested refinement of quadrilateral and hexahedral finite elements 563 Note that the faces of K may not be planar as they are images of non-linear mappings (Q 1 mappings.) We list some coefficients so that we can see later how the coefficients of higher order terms would go to zero at higher orders. (6) ˆxŷ : a 110 = (v 5 + v 7 v 6 v 8 )/8 + (v 1 + v 3 v v )/8 (top and bottom faces), ˆxẑ : a 101 = (v + v 7 v 3 v 8 )/8 + (v 1 + v 6 v v 5 )/8 (left and right faces), ŷẑ : a 011 = (v + v 7 v 3 v 6 )/8 + (v 1 + v 8 v v 5 )/8 (front and back faces), ˆxŷẑ : a 111 = (v 5 + v 7 v 6 v 8 )/8 (v 1 + v 3 v v )/8 = (v + v 7 v 3 v 8 )/8 (v 1 + v 6 v v 5 )/8 = (v + v 7 v 3 v 6 )/8 (v 1 + v 8 v v 5 )/8. As we define elements by the image of Q 1 mappings and we only use the nested refinement, the only restriction on the grids would be that the vertices and the order of vertices are provided such that the Jacobian matrix and Jacobian of the inverse mapping F 1 are bounded pointwise initially on the first level, i. e., for all first level elements, K T 1, (7) J F ( ˆx) 0, := max det(t K ( ˆx)) >C 1, F 1 1, := max (T 1 K ) ij <C, F 1, := max ( T 1 K ) ij <C, for some constant 0 <C<, where T K isa or3 3 matrix consisting of all partial derivatives of mapping F. For example, the (1, ) entry of (T K ) is x/ ŷ. There are many references on the existence and boundedness of the Jacobian of F in (7), for example, page 7 in [9]. In the nested refinement, we exclude general quadrilateral elements such as, for example, the one shaped like a triangle in D (cf. [8]). We do not have, under the nested refinement, anisotropic elements (long and narrow) either (cf. [1]). In the nested refinement, the shapes of sub-elements would become regular, closer to parallelogram or parallelepiped. The higher level Q 1 mappings would converge to P 1 mappings. We will first define a measure for non-parallelism of quadrilaterals and hexahedra. Some other people defined similar (cf. [15,10]) or different (cf. [,17]) measures.

6 56 S. Zhang ˆv ˆv 3 ˆv 5 ˆv 8 top face ˆv 6 ˆK [] right face ˆv 7 ˆK [6] ˆv 1 ˆv left face ˆK [5] front face ˆK [] ˆv ˆv 1 ˆv bottom face ˆK [1] ˆv 3 back face ˆK [3] Fig.. The reference element ˆK in D or in 3D Definition.1 The irregularity of K is defined as v 1 + v 3 v + v l for D K (see Fig. 1 or.) s(k) = max 1 i 6 s(k[i] ) for 3D K, where K [i] are faces of K. (See Fig..) Remark.1 We define s(k) as the distance between the mid-points of two diagonals of a quadrilateral in D. In 3D, s(k) is the maximum of the all 6 distances between the mid-points of diagonals on the six faces. Proposition.1 The irregularity of K is zero, i.e., s(k) = 0, if and only if K is a parallelogram in D, or a parallelepiped in 3D. Proof. First we consider a -vertex K (see Fig. 1.) If s(k) = 0, v v 1 = v ( v 1 v1 + v 3 + v ) + v = v 3 v. It means v 1 v = v v 3 and that v 1 v v 3 v is a (planar) parallelogram. The reverse is obvious and well known. Now for an 8-vertex K, ifs(k) is zero, then all 6 faces are planar and all are parallelograms as we just proved. Therefore, K is a parallelepiped. Again, the reverse is trivial in 3D. As we remarked earlier, the faces of a 3D K may not be flat (see Fig. 3.) However all its edges are straight. Proposition. For both a -vertex and an 8-vertex K, all its edges are straight line segments.

7 On the nested refinement of quadrilateral and hexahedral finite elements 565 Proof. When restricting to an edge the mapping F from the reference element ˆK to K, (or just examining the nodal basis functions on K,) we get an affine mapping which maps this edge of ˆK to a straight line segment. The two end points of this line segment are two vertices of K. In order to define our half-sizing nested refinement, we introduce the middle points. The mid-point of an edge v 1 v would be (v 1 + v )/. What do we mean by the mid-point of a -vertex quadrilateral (may not be flat), or a 8-vertex hexahedron? We define them similarly by the averages of coordinates of vertices, 1 N N v i, N = in D, or 8 in 3D. i=1 Proposition.3 For the Q 1 mapping F(ˆv) from the reference element ˆK to K, it maps the mid-points of edges, faces and solid of ˆK to the corresponding mid-points of K. Proof. We will prove the 3D case only. The D case would be proved the same way. First, we examine the element-mapping F which is defined by the vertices of K. By () (5), the mid-point of ˆv 1 ˆv (see Fig. ), i. e., (0, 1, 1), is mapped by F to v v 8 + v 30 + v 0 + v v v v 8 0 = v 1 + v. This shows all mid-edge points are mapped to mid-points of edges. The center of a face, for example, face ˆv 1 ˆv ˆv 3 ˆv, i. e., (0, 0, 1) is mapped by F to v v 8 + v v 8 + v 50 + v v v 8 0 = v 1 + v + v 3 + v, i. e., a mid-point on a face of K. Finally, the center of ˆK, i. e., (0, 0, 0) is mapped to v 1 + v + v 3 + v + v 5 + v 6 + v 7 + v 8, 8 the center of K. One step of the nested refinement of the reference element ˆK in D or 3D is done by subdividing it into or 8 sub-elements by the coordinate axes or planes, respectively, i. e., ˆK = (I 1 I 1 ) (I I 1 ) (I I ) (I 1 I ) in D, ˆK = (I 1 I 1 I 1 ) (I I 1 I 1 ) (I I I 1 ) (I 1 I I 1 ) (I 1 I 1 I ) (I I 1 I ) (I I I ) (I 1 I I ) in 3D,

8 566 S. Zhang where I 1 = [ 1, 0] and I = [0, 1]. Thanks to Proposition.3, we can define one step of the nested refinement of any element K by Q 1 mappings: Definition. One step of the nested refinement of K with an associated reference mapping F is defined by the images of vertices of sub-elements of ˆK under the mapping F. We note that the refinement is defined by the vertices of sub-elements. Once vertices of a sub-element K s are known, there is a unique mapping F s which maps the vertices of reference element to the given vertices. Then K s is defined by F s. Before a refinement, there is another mapping F which maps the reference element to the big (father) element K, from which K s was cut out. Here one would have a few questions whether Definition. is well-defined. Whether the boundaries of K and K s defined by K and K s respectively do match? Will the or 8 sub-elements K s of K together really form K? Are there any gaps or overlaps between sub-elements of K? The key is that when restricted on an interface between two elements, the Q 1 mappings are identical, determined only by the vertices of the interface. This is a corollary of the Lemma 3.1 later. Similarly we can easily show that the initial mesh T 1 would really define a partition of even if the mesh has curved-face elements. We can recursively apply Definition. to each of the new sub-elements, to define elements of higher levels of the nested refinement. Definition.3 The level-(k + 1) nested refinement of K is defined by applying one step of the nested refinement to each kth level sub-element of K, K i,k, 1 i n(k 1). Here K = K 1,1. We will show that the sub-elements converge to parallelograms or parallelepiped rapidly in the nested refinement. Theorem.1 Let K s be one of the or 8 sub-elements nestedly refined from K. (8) s(k s ) s(k). Proof. By symmetry, we only need to compute s(k s ) for the sub-element having v 1 as a vertex (see Figs. 1 and ). In D (K is not necessarily planar) s(k s ) is the distance between the two midpoints of two diagonal lines: (9) s(k s ) = v s,1 + v s,3 v s, + v s, l = 5v 1 + v + v 3 + v 8 = v 1 v + v 3 v l 8 = 1 s(k). v 1 + v + v l

9 On the nested refinement of quadrilateral and hexahedral finite elements 567 For a 3D element K, again let K s be the sub-element having v 1 as a vertex. For the three outside faces (see Fig. for face numbering,) we can apply (9) to get three equalities: (10) s(k s [i] ) = 1 s(k[i] ) 1 s(k), for i = 1, 3, 5. By symmetry, we only need to check one of the other three internal faces. We pick the top face, K [] : s(k [] s ) = v s,5 + v s,7 v s,6 + v s,8 l = 5v 5 + 5v 1 + v + v 3 + v + v 6 + v 7 + v 8 16 v 5 + v 1 + v + v 6 + v + v 8 l ( 8 ( v1 + v 3 v5 + v 7 = 1 8 ( 1 8 v 1 + v 3 v + v v + v ) + + l v 5 + v 7 v ) 6 + v 8 v 6 + v 8 l l ) = 1 s(k [1] ) + s(k [] (11) ) 1 s(k). The proof is done as all six faces have the bound (8) by (10) and (11). Therefore for the nested refinement, the sub-elements are about the 1/ size of the original element, but the irregularity of each sub-element is only 1/ or less of that of the original element. The estimates in Theorem.1 would indicate the orders of high order coefficients (3) and (6) are even higher, approaching 0 when h k 0, as we shall see in Section. 3 Convergence of Q 1 elements under the nested refinement The theory for D and 3D quadrilateral elements is well established (see [6]). Since we use only nested-refinement grids, the proof of theory would be nearly trivial. In this section, we will present the theory in this special circumstance. Lemma 3.1 Under the nested refinement, the Q 1 mappings for K i,k (see Definition.3 for notations) on level k, F i,k, are uniform scalings of the restrictions of the original Q 1 mapping from ˆK to K = K 1,1, F = F 1,1, with shifts. That is,

10 568 S. Zhang (1) F 1 F i,k : ˆK ˆK, ˆx i,k + k+1 ( ˆx + 1) (13) F 1 F i,k ( ˆx,ŷ,ẑ) = ŷ i,k + k+1 (ŷ + 1), ẑ i,k + k+1 (ẑ + 1) where ( ˆx i,k, ŷ i,k, ẑ i,k ) is the first vertex of the i-th sub-cube/square on the k-th level of ˆK. Proof. As we can repeatedly compose mappings F 1 j,k 1 F i,k for consecutive level numbers k, we only need to prove the case k =, i.e., for one level of refinement. Further, we need only to prove the case i = 1, i.e., for K 1,, the corner sub-quadrilateral or sub-hexahedron having the original vertex v 1.For the other 3 (D) or 7 (3D) sub-elements at the other corners, we can use an auxiliary rotation mapping from the reference element to itself (composed with F ) and its inverse (composed with F i,k ) in the proof. Let us denote K 1, as K s. For simplicity of notation, we only prove the D case. When restricting F (for K, i.e., F 1,1 ) on each horizontal or vertical line, it is an affine mapping (noting the special coordinates of the vertices of the reference element ˆK): F ( β[α ˆv 1 + (1 α)ˆv ] + (1 β)[α ˆv + (1 α)ˆv 3 ] ) = βαf(ˆv 1 ) + β(1 α)f(ˆv ) + (1 β)αf(ˆv ) + (1 β)(1 α)f(ˆv 3 ) = βαv 1 + β(1 α)v + (1 β)αv + (1 β)(1 α)v 3, where 0 α, β 1. Restricted on the first sub-element K s,wehave0 α, β 1/. Considering the reference mapping from ˆK to K s,wehave ( F s βs [α s ˆv 1 + (1 α s ) ˆv ] + (1 β s )[α s ˆv + (1 α s ) ˆv 3 ] ) = β s α s F s ( ˆv 1 ) + β s (1 α s )F s ( ˆv ) + (1 β s )α s F s ( ˆv ) +(1 β s )(1 α s )F s ( ˆv 3 ) = β s α s v 1 + β s (1 α s ) v 1 + v +(1 β s )α s v 1 + v = (1 + β s)(1 + α s ) + (1 β s)(1 α s ) v 3. + (1 β s )(1 α s ) v 1 + v + v 3 + v v 1 + (1 + β s)(1 α s ) v + (1 β s)(1 + α s ) v To find the composition of the two mappings, F 1 and F s, we only need to identify the relations between the parameter pairs, α, β, α s, and β s : α = 1 + α s, β = 1 + β s.

11 On the nested refinement of quadrilateral and hexahedral finite elements 569 Finally, for any point ( ˆx,ŷ) on ˆK,wehave Therefore, α s = 1 ˆx, β s = 1 ŷ. ( ˆx,ŷ) F 1 F s β[α ˆv1 + (1 α)ˆv ] + (1 β)[α ˆv + (1 α)ˆv 3 ] = (1 + β s)(1 + α s ) + (1 β s)(1 + α s ) = ( α s, β s ) = ( ˆx 1, ŷ 1 ). ˆv 1 + (1 + β s)(1 α s ) ˆv ˆv + (1 β s)(1 α s ) ˆv 3 Lemma 3. Let F i,k be the Q 1 mapping for a kth level sub-element K i,k of K and J i,k the Jacobian (see Definition.3 and (.3.1) (.3.15) of [6] for notations). J i,k 0,,Ki,k n(k 1) J 1,1 0,,K, F i,k l,, ˆK l(k 1) F 1,1 l,, ˆK, l = 1,, 3, F 1 i,k 1,, ˆK k 1 F 1 1,1 1,, ˆK, F 1 i,k,, ˆK k 1 F 1 1,1,, ˆK. Proof. Because of Lemma 3.1, we have the following scalings for the partial derivatives: F 1 F i,k ( ˆx,ŷ,ẑ) = k+1. ˆx Expressing all the partial derivatives in matrices, we would get [ ] F 1 [ ] 1 [ i,k Fi,k F = = k 1 1 ] [ ] F 1 = k 1 1,1. x ˆx x x In another word, the Jacobian and Jacobian matrices for K i,k are the same as that for K (defined on a smaller region) after factoring out a constant. The first three inequalities are proven. Further if we restrict the norms on the right hand sides of these three inequalities to proper subdomains, the inequalities would become equalities. The last inequality is implied by the following formula (cf. page 0 of [6]): F 1 i,k,, ˆK F 1 i,k,, ˆK Fi,k 3 1,, ˆK.

12 570 S. Zhang For a D triangle, nested refinement would produce similar sub-triangles whose Jacobian and Jacobian matrix are all the same after scaling. This is no longer true in 3D. For example, under the correct half-sizing refinement, a regular tetrahedron would produce two types of tetrahedra in the sequence of nested refinement (cf. for example, [18].) Other than parallelograms and parallelepipeds, -node and 8-node elements would produce infinity many new types of sub-elements under the nested refinement. By Lemma 3., all sub-elements are of better shapes as they all tend to parallelograms or parallelepipeds. For such nested-refinement meshes, as they are locally nearly uniform, they tend to produce smoother local errors and could provide better approximation for the finite element solutions, even superconvergence (cf. [16]). Nevertheless such nested-refinement meshes are not necessary for the finite element solutions to converge in Theorem 3.1 below. The theorem holds for general, irregular, but quasiuniform meshes. Now we apply the nested Q 1 elements in solving the following nd order self-adjoint elliptic problem: i (a i,j j u) = f in, i,j u = 0 on, where (a i,j ) is a symmetric matrix over (a bounded polygonal domain in D or 3D), and is uniformly positive definite on the domain. We are given an initial grid T 1 ={K} on and the nested (refined) grids, T k, k=1,,.... The finite element spaces of continuous, piece-wise Q 1 elements (on the reference elements) are defined as V k = { u h H 1 0 ( ) u h K F Q 1 ( ˆK), for K T k, and F : ˆK K Then the Galerkin finite element approximations are defined: Find u h V k, such that }. (1) a(u h,w)= (f, w) w V k, where a(u h,w) = ai,j j u h i w and (, ) is the L -inner product over. Theorem 3.1 Under the assumption (7) for all finite elements in the initial triangulation of, the Q 1 finite element solutions of nd order elliptic problem (1) converge at the optimal order: u u h 0 + h k u u h 1 Ch k u. Proof. Thanks to the nested refinement and Lemma 3., the theorem is proved by the general theory of iso-parametric elements in [6].

13 On the nested refinement of quadrilateral and hexahedral finite elements 571 Approximation of Q 1 mappings by affine mappings When computing the finite element solution of (1), one needs to compute or approximate the integrals involving rational functions, instead of typical polynomials. It was investigated by many numerical analysts how to integrate rational functions analytically, cf. [11 1]. We take a simple approach in this paper that Q 1 mappings will be replaced by affine mappings so that the Jacobian and Jacobian matrix on each element would be constant. We will analyze the perturbation by such an approximation under the nested refinement condition, and we will show that the optimal order of convergence of the finite element solutions is retained. For the Q 1 mapping F from the reference element ˆK to K, we will drop the second order term in D, or drop both the second order terms and the third order term in 3D, to get an affine mapping as follows (see (3) and (6). ) F = v i b i = a 0 + a 10 ˆx + a 01 ŷ + a 11 ˆxŷ, i=1 (15) F = a 0 + a 10 ˆx + a 01 ŷ, 8 F = v i b i i=1 F δ = a 11 ˆxŷ. = a 0 + a 100 ˆx + a 010 ŷ + a 001 ẑ + a 110 ˆxŷ + a 101 ˆxẑ + a 011 ŷẑ + a 111 ˆxŷẑ, (16) F = a 0 + a 100 ˆx + a 010 ŷ + a 001 ẑ, (17) F δ = a 110 ˆxŷ + a 101 ˆxẑ + a 011 ŷẑ + a 111 ˆxŷẑ. Let T = T + T δ be the Jacobian matrix ( F 1 / x), where T and T δ are the Jacobian matrix of the affine mapping and the perturbation, respectively: (18) ( ) F 1 i T =, T δ = T T. x or 3 3 Here x is temporarily used for either x or y in D, or for x, y or z in 3D, and the index i (i-th component of the mapping) runs from 1 to, or 3, depending the space dimension. Please note that T δ ( ) F 1 δ. x

14 57 S. Zhang We first analyze the perturbation in replacing the Q 1 mapping F by an affine mapping F in bilinear form a(, ),onthek-th level. a(u, w) = K T k a k (u, w) = K T k (19) K ( w x )T A( u x )dx = ( ŵ K T ˆK ˆx )T T T ÂT ( û ˆx )J K d ˆx, k ( ŵ ˆK ˆx )T T T Â T( û ˆx ) J K d ˆx = a(u, w) + K T k + K T k ( ŵ ˆK ( ŵ ˆK ˆx )T T T ÂT ( û ˆx )( J K J K )dˆx ( ˆx )T Tδ T ) ÂT T T ÂT δ + Tδ T ÂT δ ( û ˆx ) J K d ˆx. Here and later, d ˆx or dx in integrals stands for dxdydz or d ˆxdŷdẑ, respectively. Also the partial derivative ( w/ x) standards for the vector of two or three partial derivatives. We will estimate the four perturbations above. The first one is the Jacobian determinant perturbation: J K J K = J K [det(t ( ) F i ) 1] = J K [det(i T ˆx ( (Fδ ) i ˆx ) ) 1]. We note that K is a k-th level element and F is its reference mapping. In D, the perturbation is a rank 1 matrix (see (3)), and we get, from the well-known formula that ( (Fδ ) i det(i T ˆx det(i + ab T ) = 1 + b T a for any vectors a and b, ) ( ) = det I T ((ŷ )[ ] v1,k + v 3,k v,k v T ) ),k ˆx [ ] v1,k + v 3,k v,k v T (ŷ,k = 1 T ˆx [ ] v1,k +v When we bring the matrix T and the vector 3,k v,k v,k to level 1, by Theorem.1 and Lemma 3.1, we get, in D, [ ] (ŷ J K J K 0,, ˆK = J K v1 + v 3 v v 0,, ˆK k T 1,1 (0) C k J K 0,, ˆK. k+1 ). ˆx) 0,, ˆK

15 On the nested refinement of quadrilateral and hexahedral finite elements 573 In 3D, the perturbation is no longer a rank-one matrix. Instead we need to estimate the determinant of this matrix, ŷ ẑ 0 ẑŷ I T ˆx a110 T + 0 a101 T + ẑ a011 T + ˆxẑ a T 111 =: I B. 0 ˆx ŷ ˆxŷ But by the relation between eigenvalues and determinants, and by the relations between eigenvalues and induced matrix norms, we have (1 B ) 3 det(i B) (1 + B ) 3. Here the double bars in = 0,, ˆK denote a matrix norm, while previously the notation 0,, ˆK denotes a function norm. Again, by Theorem.1 and Lemma 3.1, we have and consequently (1) B C 1 k, J K J K 0,, ˆK J K 0,, ˆK det(i B) 1 C k J K 0,, ˆK, where C depends on the level one reference mappings only, i.e., depends only on the initial grid. Therefore (0) holds in 3D case as well. The second, the third and the fourth perturbations in (19) would all involve the analysis of ( ) 1 ( F ) T δ T 1 = (T T)T 1 F = I ( = I I T ( Fδ ˆx ˆx ˆx )) 1 ( = I T ( )) 1 ( Fδ T ˆx ( )) Fδ. ˆx Note again that T δ ( F δ / ˆx) 1.We can bring the product to level one again, as above, by Theorem.1 and Lemma 3.1. Noting that F δ contains nd-order and 3rd-order terms where the coefficients are the differences used in defining the irregularity of elements in Definition.1 and estimated in (8), we get T δ T 1 0,,K T( F δ) 0,,K () C k 1 T( F δ ) 0,,K 1 C C 1 k, k where C and C 1 depend only on the initial grid. Lemma.1 For grid level k high enough, a k (, ) is V -elliptic, i.e., where γ>0 is independent of k. a k (u, u) γ a(u, u), u V k,

16 57 S. Zhang Proof. Because of the positiveness of ( û ˆx )T T T Â T( û ), by (0), we get ˆx a k (u, u) (1 C k ) ( û K T ˆK ˆx )T T T Â T( û ˆx )J K d ˆx. k Next, we rewrite T T Â T { } = T T ( TT 1 ) T Â( TT 1 ) T and apply () to get the following estimates for the singular value of matrix TT 1 at any point on the reference element ˆK: σ j ( TT 1) = σ j ( I Tδ T 1) 1 C k. Therefore, we complete the proof as a k (u, u) (1 C k ) 3 σ min (Â) K T k σ max (Â) C 1 (1 C k )a(u, u). ( û ˆK ˆx )T T T ÂT ( û ˆx )J K d ˆx Lemma.1 ensures the existence of affine-mapping finite element solutions: Find u k V k such that (3) a k (u k,w)= (f, w) k w V k, where (f, w) k is the L -inner product with Q 1 Jacobians being replaced by affine Jacobians each element, as in the definition of a k (, ). We remark that the restriction on level number k in Lemma.1 can be easily dropped, if we provide some more details here. We will show next that the convergence order for u k would be optimal, i.e., the same as that for the regular finite-element solutions u h defined in the last section. Lemma. For grid level k high enough, it holds that (f, w) (f, w) k C k f 0 w 0 w V k. Proof. By (0) (1), we can easily prove the lemma, (f, w) (f, w) k = (f F )(w F)J(J 1 J 1)dˆx K T ˆK k J max K T k J 1 f F w F J dˆx C k f 0 w 0 0,, ˆK K T k ˆK

17 On the nested refinement of quadrilateral and hexahedral finite elements 575 Lemma.3 For grid level k high enough, it holds that a(u, w) a(u, w) k C k u 1 w 1 u, w V k. Proof. For any u, w V k, a k (u, w) a(u, w) ( ŵ K T ˆK ˆx )T T T ÂT ( û ˆx )( J K J K )dˆx k + ( ŵ ( ) K T ˆK ˆx )T T T  ( TT 1 ) T Â( TT 1 ) k T( û ˆx ) J K d ˆx. We remark that the second term here is written differently from that in (19) just for the purpose of emphasizing a symmetric matrix there. (19) shows better the small perturbations when substituting rational-function Jacobian matrices by constant matrices. But in the proof below, the second term is estimated by a decomposition of (19). The first term is the perturbation of the Jacobian determinant, and is estimated in (0) (1): () ( ŵ K T ˆK ˆx )T T T ÂT ( û ˆx )( J K J K )dˆx C k u V w V k C 1 k u 1 w 1. Here V is the energy norm defined by a(, ). In the second term, the part of perturbation by J K is treated just like the line above, (). We consider the symmetric matrix  ( TT 1 ) T Â( TT 1 ) 0,, ˆK = (T δt 1 ) T  + Â(T δ T 1 ) +(T δ T 1 ) T Â(T δ T 1 ) 0,, ˆK C 1  0,, ˆK ( T δt 1 0,, ˆK + T δ T 1 0,, ˆK ) C k A 0,, C 3 k. Applying the Cauchy-Schwartz inequality, we complete the proof of the theorem, by giving the following estimate for the second term above.

18 576 S. Zhang ( ŵ ( ) K T ˆK ˆx )T T T Â ( TT 1 ) T Â( TT 1 ) T( û ˆx ) J K d ˆx k (T ŵ ˆx )T Â ( TT 1 ) T Â( TT 1 ) T û ˆx J K d ˆx K T k C 3 k K T k ˆK C 3 k u 1 w 1. ˆK (T ŵ ˆx )T T û ˆx J K d ˆx Theorem.1 The affine-mapping finite element solutions of (3) converge at the optimal order: u u k V Ch k u. Proof. The proof is standard by the first Strang lemma (cf. Theorem.1.1 in [6]): { } a(z k,w k ) a k (z k,w k ) u u k V C inf u z k V + sup z k V k w k V k w k V (f, w k ) (f, w k ) k +C sup w k V k w k V By Lemma. and Lemma.3, the theorem is proved. 5 Numerical tests In this section, we will show two numerical tests, a D one and a 3D one. In the D case, we pick a domain shown in Fig. 5, where the grid level is 3. We simply solve the Poisson equation with zero Dirichlet boundary condition: u xx u yy = f (5) u = 0 We pick an exact solution in, on. u(x, y) = 100(5x y)(1 x)(5y x)(1 y), which would define the right hand side function f(x,y). We use a traditional method and the new method of using affine mappings in computation. The maximal nodal errors are listed in Table 1. In the left column, we computed the bilinear form a(u, w) by the -node quadrature formula provided in [7], and we computed the right hand side functional

19 On the nested refinement of quadrilateral and hexahedral finite elements 577 (0., 1) (1, 1) (1, 0.) (0, 0) Fig. 5. Nested refinement of a quadrilateral Table 1. Nodal errors of D finite element solutions level u u h l u u k l (f, w) exactly. In the right column, we used approximate affine mappings (i.e., dropping the high order terms in Q 1 reference mappings) in computing both a(u, w) and (f, w). The data show that the new method keeps the optimal order convergence, i.e, O(h ). To be exact, in both Tables 1 and, the data show the O(h ) convergence of nodal errors, which implies immediately at least the O(h ) convergence in L norm. Here we do get only the O(h ) convergence in L. But for simplicity, we list only the errors in l norm. In our 3D computation, we altered the top plane of cube to get our domain of computation, shown in Fig. 6. In Fig. 6, on the right, we draw the image of F, which is obtained by dropping the higher order terms in the Q 1 reference mapping from ˆK to K. In D, after dropping higher order terms in Q 1 mapping to get affine mappings, the resulting mapped elements (parallelograms) would have the same area as the original quadrilateral, as proved in [3]. In 3D, it is no longer true. The two hexahedra in Fig. 6 do not have the same volume. We also solve the Poisson equation (5) in 3D. Again we pick an exact solution u(x,y,z)= 6 x(1 x)y(1 y)z(1 z x y ),

20 578 S. Zhang Fig. 6. A hexahedron K and a parallelepiped K (mapped by the corresponding affine mapping) Table. Nodal errors of 3D finite element solutions level u u h l u u k l which then defines the right hand side f. In the left column of Table, we used the -points product Gauss quadrature formula on the reference cube to approximate bilinear forms a(u, v) and (f, v). The right column was done by using affine mappings. The maximal nodal errors are listed in Table which shows the optimal order of convergence for both cases, i. e., nodal errors are reduced by about a factor 1/ each level where the grid size is reduced by 1/. The mathematical definitions of affine-mappings in (15) (16) are precise, i.e., all higher order terms in reference mappings are dropped. Consequently the constant Jacobian matrix T in (18) is well-defined. This is how the author implemented the algorithm in the numerical computation. The constant Jacobian matrix T was evaluated quickly by a combination of the coordinates of or 8 vertices of the element with weights 0, or ±1. However, the new method is almost identical to the one-point quadrature formula at the center of the element (cf. Lemma 3.1), in the sense that in the new method T is evaluated at one-point (becoming a constant matrix) while the bilinear functions in the integration are still evaluated exactly. In another word, T is T at the center of the element. But in computation, we do not compute T at all, as it takes as many arithmetic operations to find only the coordinates of the center of an element as to find T itself.

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