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 con- vergence estimates are obtained in terms of weighted Sobolev spaces, and numerical results are presented.

Committee Chair

Ari Stern

Committee Members

Renato Feres

Degree

Doctor of Philosophy (PhD)

Author's Department

Mathematics

Author's School

Graduate School of Arts and Sciences

Document Type

Dissertation

Date of Award

Spring 5-15-2022

Language

English (en)

Author's ORCID

http://orcid.org/0000-0002-7829-8017

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Mathematics Commons

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