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On the nested refinement of quadrilateral and hexahedral finite elements and the affine approximation. (English) Zbl 1065.65135
This interesting and useful paper is devoted to the theory of finite element methods dealing with planar quadrilateral and three-dimensional “hexahedral” elements (cells of the grid). These quasicube cells are the images of the reference cube \([-1,1]^n,\;n=2,3\) under \(n\)-linear transformations that can be determined by the images of vertices (in the given order). It is important that the images of edges are straight line segments, for \(n=3\) the images of faces can be nonplanar. If \(n=2\) then a quasicube is a strictly convex quadrilateral.
The author defines a measure of nonregularity of the cell that corresponds to its distinction from a parallelogram or a parallelepiped. He proves that under proper refinement procedures the introduced measure tends to zero. It enables him to replace \(n\)-linear transformations by affine ones and deal only with piecewise constant approximations for Jacobians. So the computational work for getting the corresponding linear system is essentially simplified without noticeable loss of the accuracy of the modified grid method. Numerical examples are given.

MSC:
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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[1] Apel, T.: Anisotropic interpolation error estimates for isoparametric quadrilateral finite elements. Computing 60(2), 157–174 (1998) · Zbl 0897.65003 · doi:10.1007/BF02684363
[2] Arnold, D. N., Boffi, D., Falk, R. S.: Approximation by quadrilateral finite elements. Math. Comp. 71(239), 909–922 (2002) · Zbl 0993.65125 · doi:10.1090/S0025-5718-02-01439-4
[3] Arunakirinathar, K., Reddy, B. D.: Some geometrical results and estimates for quadrilateral finite elements. Comput. Methods Appl. Mech. Engrg. 122(3–4), 307–314 (1995) · Zbl 0846.65058
[4] Arunakirinathar, K., Reddy, B. D.: Further results for enhanced strain methods with isoparametric elements. Comput. Methods Appl. Mech. Engrg. 127(1–4), 127–143 (1995) · Zbl 0862.73056
[5] Arunakirinathar, K., Reddy, B. D.: A stable affine-approximate finite element method. SIAM J. Numer. Anal. 40(1), 180–197 (2002) · Zbl 1215.74077 · doi:10.1137/S0036142900382442
[6] Ciarlet, P. G.: The finite element method for elliptic problems. North-Holland, Amsterdam, 1978 · Zbl 0383.65058
[7] Girault, V.: A local projection operator for quadrilateral finite elements. Math. Comp. 64, 1421–1431 (1995) · Zbl 0856.65128 · doi:10.1090/S0025-5718-1995-1308454-X
[8] Jamet, P.: Estimation of the interpolation error for quadrilateral finite elements which can degenerate into triangles. SIAM J. Numer. Anal. 14(5), 925–930 (1977) · Zbl 0383.65009 · doi:10.1137/0714062
[9] Krizek, M., Neittaanmaki, P.: Finite element approximation of variational problems and applications. Pitman Monographs and Surveys in Pure and Applied Mathematics 50, Longman, Harlow, 1990
[10] Ming, P., Shi, Z.: Quadrilateral mesh. Chin. Ann. Math., Ser. B 23(2), 235–252 (2002) · Zbl 1005.65131 · doi:10.1142/S0252959902000237
[11] Mizukami, A.: Some integration formulas for a four-node isoparametric element. Comput. Methods Appl. Mech. Engrg. 59(1), 111–121 (1986) · Zbl 0595.73079 · doi:10.1016/0045-7825(86)90027-7
[12] Okabe, M.: Analytical integral formulae related to convex quadrilateral finite elements. Comput. Methods Appl. Mech. Engrg. 29(2), 201–218 (1981) · Zbl 0469.65011 · doi:10.1016/0045-7825(81)90113-4
[13] Rathod, H. T.: Some analytical integration formulae for a four node isoparametric element. Comput. & Structures 30(5), 1101–1109 (1988) · Zbl 0678.73045 · doi:10.1016/0045-7949(88)90153-8
[14] Rathod, H. T., Islam, Md. Shafiqul: Some pre-computed universal numeric arrays for linear convex quadrilateral finite elements. Finite Elem. Anal. Des. 38(2), 113–136 (2001) 74S05 · Zbl 0987.65115 · doi:10.1016/S0168-874X(01)00053-1
[15] Shi, Z.: A convergence condition for the quadrilateral Wilson element. Numer. Math. 44, 349–361 (1984) · Zbl 0581.65008 · doi:10.1007/BF01405567
[16] Wahlbin, L. B.: Superconvergence in Galerkin finite element methods. Lecture Notes in Mathematics 1605, Springer, Berlin, 1995 · Zbl 0826.65092
[17] Yuan, K. Y., Huang, Y. S., Yang, H. T., Pian, T. H. H.: The inverse mapping and distortion measures for 8-node hexahedral isoparametric elements. J. Comput. Mech. 14(2), 189–199 (1994) · Zbl 0807.73069 · doi:10.1007/BF00350284
[18] Zhang, S.: Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes. Houston J. of Math., 21, 541–556 (1995) · Zbl 0855.65124
[19] Zhang, Z.: Personal communication on nested refinement of quadrilaterals, 1990
[20] Zhang, Z.: Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J. Numer. Anal. 34, 640–663 (1997) · Zbl 0870.73074 · doi:10.1137/S0036142995282492
[21] Zlámal, M.: Superconvergence and reduced integration in the finite element method. Math. Comp. 32, 663–685 (1978) · Zbl 0448.65068 · doi:10.2307/2006479
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