Optimally accurate higher-order finite element methods on polytopial approximations of domains with smooth boundaries

TL;DR

This paper introduces higher-order finite element methods that maintain optimal accuracy on polytopial domain approximations of smooth boundary regions, overcoming geometric representation errors.

Contribution

The authors develop a stable, higher-order finite element approach that weakly enforces boundary conditions on polytopial approximations, ensuring optimal convergence rates.

Findings

Achieves optimal $H^1$ and $L^2$ convergence rates

Maintains stability on polytopial domain approximations

Numerical examples confirm theoretical accuracy results

Abstract

Meshing of geometric domains having curved boundaries by affine simplices produces a polytopial approximation of those domains. The resulting error in the representation of the domain limits the accuracy of finite element methods based on such meshes. On the other hand, the simplicity of affine meshes makes them a desirable modeling tool in many applications. In this paper, we develop and analyze higher-order accurate finite element methods that remain stable and optimally accurate on polytopial approximations of domains with smooth boundaries. This is achieved by constraining a judiciously chosen extension of the finite element solution on the polytopial domain to weakly match the prescribed boundary condition on the true geometric boundary. We provide numerical examples that highlight key properties of the new method and that illustrate the optimal $H^1$ and $L^2$-norm convergence rates.