Applications of FEM with Macro-Elements: Discrete Elasticity Sequences and Convergence of Lagrange Elements for a Maxwell Eigenvalue Problem

Gong, Sining

Full Metadata

Overview

Year:: 2023
Contributor:: Gong, Sining (creator); Guzmán, Johnny (Advisor); Shu, Chi-Wang (Reader); Neilan, Michael (Reader); Brown University. Department of Applied Mathematics (sponsor)
Genre:: theses
Subject:: Finite element method
Extent:: xiii, 145 p.

Files

Description

Abstract:: In this thesis, we investigate two applications of finite element methods that employ macro-elements: discrete elasticity sequences and convergence of Lagrange elements for a Maxwell eigenvalue problem. We employ the combination of finite element exterior calculus and spline theories as our primary tool. Specifically, we prove that Worsey-Farin refinements inherit the shape regularity of their parent triangulations, and due to the special structure of Worsey-Farin refinements and the existence of smoother differential sequences, we investigate the convergence of the Maxwell eigenvalue problem using quadratic or higher Lagrange finite elements on Worsey-Farin splits. To this end, we construct two Fortin-like operators to demonstrate uniform convergence of the corresponding source problem. We provide numerical experiments to validate our theoretical results. Moreover, we develop conforming finite element elasticity complexes on Worsey-Farin splits in three dimensions. These complexes connect spaces for displacement, strain, stress, and load through differential operators that represent deformation, incompatibility, and divergence. We exhibit corresponding finite element spaces on Worsey-Farin meshes and develop unisolvent degrees of freedom for these finite elements. This also yields commuting (cochain) projections on smooth functions. Notably, these spaces lack extrinsic supersmoothness at subsimplices of the mesh, yet they yield the first (strongly) symmetric stress element with no vertex or edge degrees of freedom in three dimensions. Lastly, we show that the lowest order stress space uses only piecewise linear functions, which is the lowest feasible polynomial degree for the stress space.
Notes:: Thesis (Ph. D.)--Brown University, 2023

Content

Citation

Gong, Sining, "Applications of FEM with Macro-Elements: Discrete Elasticity Sequences and Convergence of Lagrange Elements for a Maxwell Eigenvalue Problem" (2023). Applied Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://repository.library.brown.edu/studio/item/bdr:quufmm5a/

Relations

Collection:

Applied Mathematics Theses and Dissertations

Theses and Dissertations for the Applied Mathematics department.
...