PhD Thesis Defense: "A nonconforming finite element method for the 2D vector Laplacian"

Abstract: The vector Laplacian presents difficulties in finite element ap- proximation. It is well known that for nonconvex domains, H1- conforming approximation spaces form a closed subspace of the solution space H(div; Ω) ∩ H ̊(curl; Ω). Hence H1-conforming approximations will fail to converge. This is problematic as it is highly difficult to construct more general finite dimensional ap- proximation spaces for this space. We will present an extension of a nonconforming method introduced by Brenner et al. The method was originally given for P1-nonconforming spaces in two dimensions. Our extension is given for degree r polynomials, but which agrees with the preceding method for the lowest degree case. The extended method is a hybridization with equivalent 1-field, 2-field, and 3-field formulations.

The regularity of the solution, and the corresponding convergence estimates are obtained in terms of weighted Sobolev spaces, and numerical results are presented.

Host: Ari Stern