Analysis of Finite Element Methods for Vector Laplacians on Surfaces

Peter Hansbo, Mats G. Larson, Karl Larsson

Published 2016-10-21Version 1

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 with one order higher polynomial degree for the mapping. The tangent condition is weakly enforced using a penalization term. We derive error estimates that takes the approximation of both the geometry of the surface and the solution to the partial differential equation into account. We also present numerical results that verify our theoretical findings.