Article

Approximation by quadrilateral finite elements

Mathematics of Computation (Impact Factor: 1.37). 06/2000; 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.

1 Bookmark
 · 
191 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: This paper focuses on the low-order nonconforming rectangular and quadrilateral finite elements approximation of incompressible flow. Beyond the previous research works, we propose a general strategy to construct the basis functions. Under several specific constraints, the optimal error estimates are obtained, i.e., the first order accuracy of the velocities in H 1-norm and the pressure in L 2-norm, as well as the second order accuracy of the velocities in L 2-norm. Besides, we clarify the differences between rectangular and quadrilateral finite element approximation. In addition, we give several examples to verify the validity of our error estimates.
    Science China Mathematics 01/2013; 56(2). · 0.50 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The unsymmetric finite element formulation has been proposed recently to improve predictions from distorted finite elements. Studies have also shown that this special formulation using parametric functions for the test functions and metric functions for the trial functions works surprisingly well because the former satisfy the continuity conditions while the latter ensure that the stress representation during finite element computation can retrieve in a best-fit manner, the actual variation of stress in the metric space. However, a question that remained was whether the unsymmetric formulation was variationally correct. Here we determine that it is not, using the simplest possible element to amplify the principles.
    Structural Engineering & Mechanics 01/2007; · 0.80 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: In this manuscript we compare physical and reference frame discontinuous Galerkin (dG) discretizations with emphasis on the influence of reference-to-physical frame mappings on the discrete space properties. We assess the excellence of physical frame discrete spaces in terms of approximation capabilities as well as the increased flexibility compared to reference frame discretizations. As a matter of fact, whenever curved elements are considered, non-affine reference-to-physical frame mappings are able to spoil the convergence properties of reference frame discrete spaces. This poorly documented drawback does not affect basis functions defined directly in the physical frame. The convergence degradation associated to reference frame discretizations is evaluated theoretically, providing error bounds for the approximation error of the L 2-orthogonal projection operator, and the findings are justified by means of numerical test cases. In particular we exemplify by means of quadrilateral elements grids challenging grid configurations characterized by non-affine mappings and demonstrate the ability to predict the convergence rates without stringent assumptions on the element shapes.
    Journal of Scientific Computing 01/2012; 52(3). · 1.71 Impact Factor

Full-text (2 Sources)

View
67 Downloads
Available from
May 31, 2014