Analysis of finite element methods for vector Laplacians on surfaces

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 that 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 into account the approximation of both the geometry of the surface and the solution to the partial differential equation. 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.

C0IPG for a Fourth Order Eigenvalue Problem

This paper concerns numerical computation of a fourth order eigenvalue problem. We first show the well-posedness of the source problem. An interior penalty discontinuous Galerkin method (C0IPG) using Lagrange elements is proposed and its convergence is studied. The method is then used to compute the eigenvalues. We show that the method is spectrally correct and prove the optimal convergence. Numerical examples are presented to validate the theory.

A viscoelastic thin rod model for large deformations: Numerical examples

We present a Cosserat-based 3D–1D dimensional reduction for a viscoelastic finite strain model. The numerical resolution of the reduced coupled minimization/evolution problem is based on a splitting method. We start by approximating the minimization problem using the finite element method with P1 Lagrange elements. The solution of this problem is used in the time-incremental formulation of the evolution problem.