Journal
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
Volume 24, Issue 8, Pages 1701-1727Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218202514400077
Keywords
Polygonal and polyhedral meshes; finite elements; patch test; quadrature error; mimetic finite differences
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Funding
- US National Science Foundation [1321661]
- Donald B. and Elizabeth M. Willett endowment at the University of Illinois at Urbana-Champaign
- Div Of Civil, Mechanical, & Manufact Inn
- Directorate For Engineering [1321661] Funding Source: National Science Foundation
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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.
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