Analysis of Finite Element Methods for Vector Laplacians on Surfaces
This work provides a rigorous numerical analysis for a class of surface PDEs relevant to fluid dynamics and elasticity, but the method is an incremental extension of existing finite element techniques.
The paper develops a finite element method for the vector Laplacian on surfaces, using continuous Lagrange elements with a penalty term to enforce tangency. Optimal-order error estimates are derived and verified numerically, showing that the normal approximation in the penalty must match the solution's approximation order.
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.