Arnold, Douglas N.; Boffi, Daniele; Falk, Richard S. Approximation by quadrilateral finite elements. (English) Zbl 0993.65125 Math. Comput. 71, No. 239, 909-922 (2002). This paper concerns the approximation properties of finite element spaces on quadilateral meshes. The finite element spaces are constructed starting with a finite-dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism on the space onto the element. It is demonstrated the degradation of the convergence order on quadrilateral meshes as compared to triangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements. Reviewer: Laura-Iulia Aniţa (Iaşi) Cited in 1 ReviewCited in 90 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65D07 Numerical computation using splines 41A10 Approximation by polynomials 41A25 Rate of convergence, degree of approximation 41A27 Inverse theorems in approximation theory 41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX) Keywords:quadrilateral finite element; mixed finite element; convergence; serendipity finite elements PDF BibTeX XML Cite \textit{D. N. Arnold} et al., Math. Comput. 71, No. 239, 909--922 (2002; Zbl 0993.65125) Full Text: DOI arXiv References: [1] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058 [2] B. Nagy, On cosine operator functions in Banach spaces, Acta Sci. Math. (Szeged) 36 (1974), 281 – 289. · Zbl 0273.47008 [3] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801 [4] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. · Zbl 0585.65077 [5] F. Kikuchi, M. Okabe, and H. Fujio, Modification of the 8-node serendipity element, Comp. Methods Appl. Mech. Engrg. 179 (1999), 91-109. · Zbl 0979.74067 [6] R. H. McNeal and R. L. Harder, Eight nodes or nine?, Int. J. Numer. Methods Engrg. 33 (1992), 1049-1058. [7] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differential Equations 8 (1992), no. 2, 97 – 111. · Zbl 0742.76051 · doi:10.1002/num.1690080202 · doi.org [8] P. Sharpov and Y. Iordanov, Numerical solution of Stokes equations with pressure and filtration boundary conditions, J. Comp. Phys. 112 (1994), 12-23. · Zbl 0798.76042 [9] G. Strang and G. Fix, A Fourier analysis of the finite element variational method, Constructive Aspects of Functional Analysis , C.I.M.E. II Ciclo, 1971, pp. 793-840. · Zbl 0272.65099 [10] Jing Zhang and Fumio Kikuchi, Interpolation error estimates of a modified 8-node serendipity finite element, Numer. Math. 85 (2000), no. 3, 503 – 524. · Zbl 0970.65119 · doi:10.1007/s002110000104 · doi.org [11] O. C. Zienkiewicz and R. L. Taylor, The finite element method, fourth edition, volume 1: Basic formulation and linear problems, McGraw-Hill, London, 1989. · Zbl 0974.76003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.