On optimal finite element schemes for biharmonic equation
Shuo Zhang
TL;DR
This paper develops two nonconforming finite element schemes of cubic and quartic order for the planar biharmonic equation, achieving optimal convergence rates on general shape-regular triangulations, thus enabling high-order solutions on complex meshes.
Contribution
It introduces the first construction of arbitrary order optimal finite element schemes for the biharmonic equation on general shape-regular triangulations.
Findings
Optimal convergence rates achieved for both schemes
Construction valid for arbitrary order elements
Applicable to general shape-regular meshes
Abstract
In this paper, two nonconforming finite element schemes that use piecewise cubic and piecewise quartic polynomials respectively are constructed for the planar biharmonic equation with optimal convergence rates on general shape-regular triangulations. Therefore, it is proved that optimal finite element schemes of arbitrary order for planar biharmonic equation can be constructed on general shape-regular triangulations.
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Taxonomy
TopicsAdvanced Numerical Methods in Computational Mathematics · Differential Equations and Numerical Methods · Advanced Mathematical Modeling in Engineering