Addressing integration error for polygonal finite elements through polynomial projections: A patch test connection

Cameron Talischi, Glaucio H. Paulino, F. Brezzi, G. Manzini

Research output: Contribution to journalArticlepeer-review

Abstract

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations.

Original languageEnglish (US)
Pages (from-to)1701-1727
Number of pages27
JournalMathematical Models and Methods in Applied Sciences
Volume24
Issue number8
DOIs
StatePublished - Jul 2014

Keywords

  • Polygonal and polyhedral meshes
  • finite elements
  • mimetic finite differences
  • patch test
  • quadrature error

ASJC Scopus subject areas

  • Modeling and Simulation
  • Applied Mathematics

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