Approximation by quadrilateral finite elements. Math. Comp. 71(239): 909-922

Mathematics of Computation (Impact Factor: 1.49). 06/2000; 71(239). DOI: 10.1090/S0025-5718-02-01439-4
Source: arXiv

ABSTRACT We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given 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 of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r+1 in L2 and order r in H1 is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

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Available from: Richard S. Falk, Nov 28, 2012
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    • "Regretfully this is not possible. Indeed from [4] we know that the accuracy of serendipity elements can be seriously deteriorated even for regular elements. The reason of that is the failure of the inclusion of P k in the interpolation space. "
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    ABSTRACT: We study $W^{1,p}$ Lagrange interpolation error estimates for general quadrilateral $\mathcal{Q}_{k}$ finite elements with $k\ge 2$. For the most standard case of $p=2$ it turns out that the constant $C$ involved in the error estimate can be bounded in terms of the minimal interior angle of the quadrilateral. Moreover, the same holds for any $p$ in the range $1\le p<3$. On the other hand, for $3\le p$ we show that $C$ also depends on the maximal interior angle. We provide some counterexamples showing that our results are sharp.
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    • "Our results here pertain to a coordinate free approach in the sense that our approximating functions u h are polynomial on each element. The case of the pullbacks being polynomials works as well, but it is more subtle to handle due to some complications including some tensor product polynomial approximation issues on quadrilateral meshes ( Arnold et al. [1]). We will report it in a separate paper. "
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    ABSTRACT: It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise polynomials, that means we have to at least maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements. In this paper we give clear definitions of numerical smoothness that address the across-interface smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in terms of differences in derivative values. Furthermore, we prove rigorously that the principle can be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons and hexahedrons in three dimensions. With this validation we can justify, among other things, incorporation of this principle in creating adaptive numerical approximation for the solution of PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential non-convergence and instability.
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    • "and let u b ∈ V b h be such thatû I + ˆ u b is the piecewise quadratic interpolation ofû in the local coordinates ξ, η. Thanks to the relation (4.35) and the interpolation theory by [2] , for τ ∈ Σ h it holds "
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    ABSTRACT: Superconvergence and a posteriori error estimators of recovery type are analyzed for the 4-node hybrid stress quadrilateral finite element method proposed by Pian and Sumihara (Int. J. Numer. Meth. Engrg., 1984, 20: 1685-1695) for linear elasticity problems. Uniform superconvergence of order $O(h^{1+\min\{\alpha,1\}})$ with respect to the Lam\'{e} constant $\lambda$ is established for both the recovered gradients of the displacement vector and the stress tensor under a mesh assumption, where $\alpha>0$ is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. A posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.
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