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Higher Order Fourier Finite Element Methods to Approximate Hodge Laplacian Problems on Axisymmetric Domains

Table of Contents

Introduction

The goal of this project is to use higher order finite element methods to construct a family of Fourier finite element spaces that can be applied to axisymmetric problems and then to apply them to discretize the Hodge Laplacian problem on axisymmetric domains. This work will extend the results of the papers by Minah Oh, de Rham complexes arising from Fourier finite element methods in axisymmetric domains, Computers & Mathematics with Applications, Volume 70, Issue 8, 2015 and The Hodge Laplacian on axisymmetric domains and its discretization, IMA Journal of Numerical Analysis, 2020, where the author uses the lowest order finite element spaces.

This is joint work between Minah Oh (Associate Professor of Mathematics at James Madison University) and Nicole Stock (Senior at James Madison Universty) as as part of Nicole Stock's Honor's Capstone Thesis.

This repository is written in conjuction with this repository, which contains the lowest order Fourier finite element methods for the same problems.

Equations

The finite element methods approximate the solution to the following weighted mixed formulation of the abstract Hodge Laplacian.

Find such that:

With .

See the next section for more details.

The four equations corresponding with are as follows:

k = 0: The Neumann Problem for the Axisymmetric Poisson Equation

See here for more

k = 1: The Axisymmetric Vector Laplacian curl curl + grad div

See here for more

k = 2: The Axisymmetric Vector Laplacian curl curl + grad div

See here for more

k = 3: The Dirichlet Problem for the Axisymmetric Poisson Equation

See here for more

Definitions

We let be defined in the following way:

and let be the Hilbert space associated with each,

Furthermore, the grad, curl, and div formulas for the n-th Fourier mode are as follows:

Requirements

  • MATLAB

James Madison University Honors Capstone Project

Author: Nicole Stock

Faculty Research Advisor: Dr. Minah Oh

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