An Adaptive Finite Element Method Based on Generalized Barycentric Coordinates

TL;DR

This paper develops an a posteriori error estimate for polygonal finite element methods using Wachspress barycentric coordinates, validating an adaptive algorithm through numerical experiments.

Contribution

It introduces a residual-based a posteriori error estimator for polygonal finite elements and proves its bounds relative to the discretization error.

Findings

The error estimator provides both upper and lower bounds for the discretization error.

Numerical experiments confirm the effectiveness of the adaptive algorithm on various domains.

Abstract

This work derives a posteriori error estimate of polygonal finite element methods based on Wachspress barycentric coordinates. In particular, we prove that the classical residual-based a posteriori error estimator is both an upper and lower bounds for the discretization error. The analysis relies a Scott-Zhang type interpolation and homogeneity arguments for rational functions on polygonal elements. Numerical experiments on square and L-shaped domains demonstrate the effectiveness of the adaptive algorithm.