Polynomial Finite Element Method for Domains Enclosed by Piecewise Conics

TL;DR

This paper develops a polynomial finite element method for curved domains bounded by piecewise conics, providing stable bases, error analysis, and efficient algorithms, with numerical results confirming its effectiveness.

Contribution

It introduces a stable Bernstein-Bézier based basis for finite elements on conic-bounded domains, along with error bounds and optimal assembly algorithms.

Findings

Numerical experiments demonstrate the method's effectiveness.

Stable local bases are constructed for curved domains.

Error bounds are established for the finite element approximation.

Abstract

We consider bivariate piecewise polynomial finite element spaces for curved domains bounded by piecewise conics satisfying homogeneous boundary conditions, construct stable local bases for them using Bernstein-B\'ezier techniques, prove error bounds and develop optimal assembly algorithms for the finite element system matrices. Numerical experiments confirm the effectiveness of the method.