Analysis of Finite Element Methods for Vector Laplacians on Surfaces

TL;DR

This paper introduces a finite element method for vector Laplacians on surfaces, accounting for geometry and solution approximation, with theoretical error estimates and numerical validation.

Contribution

It develops a surface finite element method using standard Lagrange elements with weak tangent enforcement and derives optimal error estimates considering geometry approximation.

Findings

Error estimates depend on geometry and solution approximation quality.

Higher-order elements improve accuracy and verify theoretical predictions.

Normal approximation in penalization must match solution approximation order.

Abstract

We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in $\mathbb{R}^3$. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements which describe a $\mathbb{R}^3$ vector field on the surface and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take the approximation of both the geometry of the surface and the solution to the partial differential equation into account. In particular we note that to achieve optimal order error estimates, in both energy and $L^2$ norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.