A narrow-band unfitted finite element method for elliptic PDEs posed on surfaces

TL;DR

This paper introduces a narrow-band unfitted finite element method for solving elliptic PDEs on surfaces, allowing for implicit surface representations and unaligned meshes, with proven error estimates and numerical validation.

Contribution

It presents a novel unfitted finite element approach for surface PDEs using level set representations and provides rigorous error analysis and convergence results.

Findings

Error estimates for bulk and surface solutions

Optimal convergence order in energy norm

Numerical examples confirming theoretical results

Abstract

The paper studies a method for solving elliptic partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method allows a surface to be given implicitly as a zero level of a level set function. A surface equation is extended to a narrow-band neighborhood of the surface. The resulting extended equation is a non-degenerate PDE and it is solved on a bulk mesh that is unaligned to the surface. An unfitted finite element method is used to discretize extended equations. Error estimates are proved for finite element solutions in the bulk domain and restricted to the surface. The analysis admits finite elements of a higher order and gives sufficient conditions for archiving the optimal convergence order in the energy norm. Several numerical examples illustrate the properties of the method.