Low Regularity Estimates for CutFEM Approximations of an Elliptic Problem with Mixed Boundary Conditions

TL;DR

This paper derives error estimates for a cut finite element method applied to a second order elliptic problem with mixed boundary conditions, focusing on low regularity solutions and handling boundary flux terms.

Contribution

It provides novel error estimates for CutFEM in low regularity settings, including optimal results for Dirichlet and suboptimal for mixed conditions with a logarithmic factor.

Findings

Error estimates are established for solutions with regularity s in (1, 3/2].

Optimal error bounds are achieved for Dirichlet boundary conditions.

Suboptimal bounds with a logarithmic factor are obtained for mixed Dirichlet-Neumann conditions.

Abstract

We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution $u \in H^s$ with $s\in (1,3/2]$. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.