Non-degenerate Eulerian finite element method for solving PDEs on surfaces

TL;DR

This paper introduces a non-degenerate finite element method for solving elliptic PDEs on surfaces, improving stability and convergence by extending the problem into a bulk domain and proving theoretical and numerical effectiveness.

Contribution

The paper develops a new extended formulation that results in uniformly elliptic equations in a bulk domain, enabling stable finite element solutions for surface PDEs.

Findings

The method produces convergent finite element solutions on surfaces.

Numerical examples demonstrate the stability and accuracy of the approach.

Theoretical proofs confirm convergence to the original surface PDE solutions.

Abstract

The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method builds upon the formulation introduced in Bertalmio et al., J. Comput. Phys., 174 (2001), 759--780., where a surface equation is extended to a neighborhood of the surface. The resulting degenerate PDE is then solved in one dimension higher, but can be solved on a mesh that is unaligned to the surface. We introduce another extended formulation, which leads to uniformly elliptic (non-degenerate) equations in a bulk domain containing the surface. We apply a finite element method to solve this extended PDE and prove the convergence of finite element solutions restricted to the surface to the solution of the original surface problem. Several numerical examples illustrate the properties of the method.