A Cut Discontinuous Galerkin Method for the Laplace-Beltrami Operator

TL;DR

This paper introduces a novel discontinuous Galerkin finite element method for solving the Laplace-Beltrami operator on embedded surfaces, achieving optimal error estimates and stability regardless of surface positioning.

Contribution

The paper presents a new cut finite element method using discontinuous Galerkin techniques for the Laplace-Beltrami operator on embedded surfaces, with proven stability and error bounds.

Findings

Optimal a priori error estimates derived

Condition number bounds independent of surface position

Numerical examples confirm theoretical results

Abstract

We develop a discontinuous cut finite element method (CutFEM) for the Laplace-Beltrami operator on a hypersurface embedded in $\mathbb{R}^d$. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in $\mathbb{R}^d$. The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighboring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.