Residual based Error Estimate and Quasi-Interpolation on Polygonal Meshes for High Order BEM-based FEM

TL;DR

This paper develops residual-based error estimates and quasi-interpolation techniques for high-order boundary element method-based finite element methods on polygonal meshes, enabling adaptive strategies and demonstrating optimal convergence.

Contribution

It introduces new quasi-interpolation operators and residual error estimates specifically designed for polygonal meshes in BEM-based FEM, with proven reliability and efficiency.

Findings

Residual error estimates are reliable and efficient.

Numerical experiments show optimal convergence rates.

Methods work well on non-convex and adaptively refined meshes.

Abstract

Only a few numerical methods can treat boundary value problems on polygonal and polyhedral meshes. The BEM-based Finite Element Method is one of the new discretization strategies, which make use of and benefits from the flexibility of these general meshes that incorporate hanging nodes naturally. The article in hand addresses quasi-interpolation operators for the approximation space over polygonal meshes. To prove interpolation estimates the Poincar\'e constant is bounded uniformly for patches of star-shaped elements. These results give rise to the residual based error estimate for high order BEM-based FEM and its reliability as well as its efficiency are proven. Such a posteriori error estimates can be used to gauge the approximation quality and to implement adaptive FEM strategies. Numerical experiments show optimal rates of convergence for meshes with non-convex elements on uniformly as well as on adaptively refined meshes.