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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 79
PROCEEDINGS OF THE SEVENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 134
The Finite Element Method on Quadrilateral Meshes D. Boffi
Department of Mathematics "F. Casorati", University of Pavia, Italy Full Bibliographic Reference for this paper
D. Boffi, "The Finite Element Method on Quadrilateral Meshes", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Seventh International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 134, 2004. doi:10.4203/ccp.79.134
Keywords: quadrilateral, finite elements, approximation, serendipity, mixed finite elements.
Summary
We review in a unified approach results concerning the
approximation properties of quadrilateral finite elements on general regular
(possibly nonaffine) meshes. These results have been presented recently
to the mathematical community; the main references are [1] for the scalar
case, [2] for the Stokes problem, [3] for several numerical
tests, [4] for the vector case, and [5] for possible
consequences for the Reissner-Mindlin plate. We believe that these results
are of great importance for the design of quadrilateral finite elements and the
aim of this contribution is to provide the engineering community with a unified
analysis of this topic.
We shall distinguish between the cases of scalar
and vector functions. Indeed, in the former case, we consider the standard
mapping (based on the pull-back) which preserves the function continuity. For
the latter case, we shall handle two different situations. On one side, if we
are interested in the continuity of the normal component (like in problems
arising from elasticity or acoustics, when the underlying functional space is
In the the scalar case, we are interested in optimal
approximation estimates for the function and its gradient in the
where ![]() ![]() ![]() ![]() ![]()
The extension to the vector case is not a straightforward application to the
results in the scalar case. We are interested in optimal approximation
estimates of the vectorfields and their divergences in the
Also in this case, we shall show necessary and sufficient conditions for estimates (51), (52) to hold, which characterize the finite element space on the reference element. Such conditions are a little more complicated to describe than the previous one and we do not anticipate them in this abstract.
We then present several numerical tests coming from different applications.
The first application concerns the Laplace/Poisson problem.
We show that commonly used serendipity elements on general
quadrilateral meshes present a lack of convergence. This is
the case for the 8-node element which is asymptotically second order accurate
in
Then we present the application to the popular We then show that the popular Raviart-Thomas element does not achieve optimal convergence on general quadrilateral meshes. A new mixed finite element family introduced in [4] proves to be optimal order convergent also on highly distorted meshes. Finally, we consider two eigenvalue problems. Numerical experiments confirm the theory and show that the lowest order Raviart-Thomas element provide stable, but not convergent solutions on distorted meshes. Some new numerical results show that higher order Raviart-Thomas elements provide suboptimally convergent solutions. On the other hand, the new family of finite elements introduced in [4] presents optimal convergence behavior. References
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