Quadrature for Implicitly-defined Finite Element Functions on Curvilinear Polygons

TL;DR

This paper introduces a new quadrature approach for integrating products of implicitly defined finite element functions on curvilinear polygonal meshes, simplifying computations by reducing integrals to boundary integrals.

Contribution

It presents a novel method for quadrature on curvilinear polygons that handles implicitly defined finite element functions, extending existing techniques to more complex mesh geometries.

Findings

The method effectively reduces volume integrals to boundary integrals.

Numerical experiments demonstrate practical accuracy and efficiency.

The approach is applicable to various polygonal finite element methods.

Abstract

$H^1$-conforming Galerkin methods on polygonal meshes such as VEM, BEM-FEM and Trefftz-FEM employ local finite element functions that are implicitly defined as solutions of Poisson problems having polynomial source and boundary data. Recently, such methods have been extended to allow for mesh cells that are curvilinear polygons. Such extensions present new challenges for determining suitable quadratures. We describe an approach for integrating products of these implicitly defined functions, as well as products of their gradients, that reduces integrals on cells to integrals along their boundaries. Numerical experiments illustrate the practical performance of the proposed methods.