A Trace Finite Element Method for Vector-Laplacians on Surfaces

TL;DR

This paper develops and analyzes a trace finite element method for solving vector-Laplace equations on surfaces, incorporating stabilization and preconditioning techniques, with proven error bounds and numerical validation.

Contribution

It introduces a novel trace finite element approach with stabilization and a Schur complement preconditioner for surface vector-Laplace problems.

Findings

Optimal discretization error bounds derived

Well-posed saddle point formulation established

Effective preconditioning demonstrated in numerical experiments

Abstract

We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface-independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included.