First Advisor

Jeffrey Ovall

Term of Graduation

Spring 2024

Date of Publication

5-16-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences

Department

Mathematics and Statistics

Language

English

Physical Description

1 online resource (x, 174 pages)

Abstract

We present a finite element method for linear elliptic partial differential equations on bounded planar domains that are meshed with cells that are permitted to be curvilinear and multiply connected. We employ Poisson spaces, as used in virtual element methods, consisting of globally continuous functions that locally satisfy a Poisson problem with polynomial data. This dissertation presents four peer-reviewed articles concerning both the theory and computation of using such spaces in the context of finite elements. In the first paper, we propose a Dirichlet-to-Neumann map for harmonic functions by way of computing the trace of a harmonic conjugate by numerically solving a second-kind integral equation; with the trace of a given harmonic function and its conjugate, we may obtain interior values and derivatives (such as the gradient). In the second paper, we establish some properties of a local Poisson space (i.e. when restricted to a single mesh cell), including its dimension, and provide a construction of a basis of this space. An interpolation operator for this space is introduced, and bounds on the interpolation error are proved and verified computationally in the lowest order case. In the third paper, we demonstrate that computations with higher-order spaces are computationally feasible by showing that both the H1 semi-inner product and the L2 inner product can be computed in the local Poisson space using only path integrals over boundary of the mesh cell, without need for any volumetric quadrature. Reducing the L2 inner product to a boundary integral involves determining an "anti-Laplacian" of a harmonic function, i.e. a biharmonic function whose Laplacian is given; we provide a construction of the trace and normal derivative of such a function. In the fourth paper, we show that the H1 semi-inner product and L2 inner product can be likewise computed on mesh cells that are "punctured", i.e. multiply connected. The primary difficulty arises due to the fact that a given harmonic function is not guaranteed to have a harmonic conjugate, but can be corrected for by introducing logarithmic singularities centered at chosen points in the holes. In addition to these four papers, we also provide a brief update on ongoing extensions of this work, including a full implementation of the finite element method and application to computing terms that arise in problems with advection terms and generalized diffusion operators.

Rights

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Comments

Partial funding for the research presented in this dissertation was through the following NSF grants: DMS-1414365, DMS-1522471, DMS-1624776, DMS-2012285, RTG grant DMS-2136228.

Persistent Identifier

https://archives.pdx.edu/ds/psu/42217

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