Numerical integration over implicitly defined domains for higher order unfitted finite element methods

TL;DR

This paper evaluates various numerical integration techniques over implicitly defined domains for higher order unfitted finite element methods, focusing on their accuracy, complexity, and impact on PDE solutions.

Contribution

It introduces and compares multiple integration approaches for implicit domains within unfitted FEMs, analyzing their effectiveness and computational costs.

Findings

Subdivision and moment-fitting methods improve integration accuracy.

Monte Carlo techniques offer flexible but less precise solutions.

Integration errors significantly influence FEM solution accuracy.

Abstract

The paper studies several approaches to numerical integration over a domain defined implicitly by an indicator function such as the level set function. The integration methods are based on subdivision, moment--fitting, local quasi-parametrization and Monte-Carlo techniques. As an application of these techniques, the paper addresses numerical solution of elliptic PDEs posed on domains and manifolds defined implicitly. A higher order unfitted finite element method (FEM) is assumed for the discretization. In such a method the underlying mesh is not fitted to the geometry, and hence the errors of numerical integration over curvilinear elements affect the accuracy of the finite element solution together with approximation errors. The paper studies the numerical complexity of the integration procedures and the performance of unfitted FEMs which employ these tools.