An unfitted finite element method by direct extension for elliptic problems on domains with curved boundaries and interfaces

TL;DR

This paper introduces a novel unfitted finite element method for elliptic problems on domains with curved boundaries, avoiding mesh adjustments and achieving optimal convergence without boundary-fitted meshes.

Contribution

The paper presents a new unfitted finite element approach that extends interior finite element spaces directly, imposing boundary conditions weakly, and demonstrates stability and optimal convergence.

Findings

Method achieves optimal $L^2$ and energy norm convergence.

Numerical results confirm accuracy and robustness in 2D and 3D.

No mesh adjustment or special stabilization needed.

Abstract

We propose and analyze an unfitted finite element method for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. The boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. Optimal convergence rates under the $L^2$ norm and the energy norm are derived. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.