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

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 $(\sigma , u) \in V^{k-1} x V^k$ such that:

$\begin{aligned} (\sigma , \tau)_{L^2_r(\Omega)} - (d^{k-1} \tau , u)_{L^2_r(\Omega)} &= 0, && \text{ for all } \tau \in V^{k-1}, \\ (d^{k-1}\sigma , v)_{L^2_r(\Omega)} + (d^k u, d^k v)_{L^2_r(\Omega)} &= (f,v)_{L^2_r(\Omega)} , && \text{ for all } v \in V^k. \end{aligned}$

With $k = 0,1,2,3$ .

See the next section for more details.

The four equations corresponding with $k = 0,1,2,3$ are as follows:

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

$\begin{aligned} - \text{div}^{n*}_{rz} \text{grad}^n_{rz} u & = f &&\text{ in } \Omega, \\ \text{grad}^n_{rz} u \cdot n & = 0 &&\text{ on } \Gamma_1. \end{aligned}$

See here for more

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

$\begin{aligned} - \text{grad}^n_{rz} \text{div}^{n*}_{rz} + \text{curl}^{n*}_{rz} \text{curl}^n_{rz} u &= f,\\ (\text{curl}^n_{rz} u)_{rz} \cdot t &= 0,\\ (\text{curl}^n_{rz} u)_{\theta} &= 0 &&\text{ on } \Gamma_1,\\ u_{rz} \cdot n &= 0 &&\text{ on } \Gamma_1. \end{aligned}$

See here for more

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

$\begin{aligned} \text{curl}^n_{rz} \text{curl}^{n*}_{rz} u - \text{grad}^{n*}_{rz} \text{div}^n_{rz} u &= f,\\ u_{rz} \cdot t &= 0,\\ u_{\theta} &= 0,\\ \text{div}^n_{rz} u &= 0 &&\text{ on } \Gamma_1. \end{aligned}$

See here for more

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

$\begin{aligned} - \text{div}^n_{rz} \text{grad}^{n*}_{rz} u &= f &&\text{ in } \Omega,\\ u &= 0 &&\text{ on } \Gamma_1. \end{aligned}$

See here for more

Definitions

We let $d^k$ be defined in the following way:

$\begin{align*} d^0 v &= \text{grad}^n_{rz} v,\\ d^1 v &= \text{curl}^n_{rz} v,\\ d^2 v &= \text{div}^n_{rz} v,\\ d^3 v &= 0, \end{align*}$

and let $V^k$ be the Hilbert space associated with each,

$\begin{align*} V^0 &= H_r(\text{grad}^n, \Omega),\\ V^1 &= H_r(\text{curl}^n, \Omega),\\ V^2 &= H_r(\text{div}^n, \Omega),\\ V^3 &= L^2_r(\Omega). \end{align*}$

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

$\begin{aligned} \text{grad}^{n}_{rz} v &= \left[ {\begin{array}{cc} \partial_r v \\ - \frac{n}{r} v \\ \partial_z v \end{array} } \right],\\ \text{curl}^{n}_{rz} \left[ {\begin{array}{cc} v_r \\ v_{\theta} \\ v_z \end{array} } \right] &= \left[ {\begin{array}{cc} -( \frac{n}{r} v_z - \partial_z v_{\theta} ) \\ \partial_z v_r - \partial_r v_z \\ \frac{n v_r + v_{\theta}}{r} + \partial_r v_{\theta} \end{array} } \right],\\ \text{div}^{n}_{rz} \left[ {\begin{array}{cc} v_r \\ v_{\theta} \\ v_z \end{array} } \right] &= \partial_r v_r + \frac{v_r - n v_{\theta}}{r} + \partial_z v_z . \end{aligned}$

$\begin{aligned} \text{grad}^{n*}_{rz} v &= \left[ {\begin{array}{cc} \partial_r v \\ \frac{n}{r} v \\ \partial_z v \end{array} } \right],\\ \text{curl}^{n*}_{rz} \left[ {\begin{array}{cc} v_r \\ v_{\theta} \\ v_z \end{array} } \right] &= \left[ {\begin{array}{cc} \frac{n}{r} v_z - \partial_z v_{\theta} \\ \partial_z v_r - \partial_r v_z \\ \frac{-n v_r + v_{\theta}}{r} + \partial_r v_{\theta} \end{array} } \right],\\ \text{div}^{n*}_{rz} \left[ {\begin{array}{cc} v_r \\ v_{\theta} \\ v_z \end{array} } \right] &= \partial_r v_r + \frac{v_r + n v_{\theta}}{r} + \partial_z v_z . \end{aligned}$

Requirements

MATLAB

James Madison University Honors Capstone Project

Author: Nicole Stock

Faculty Research Advisor: Dr. Minah Oh

Name		Name	Last commit message	Last commit date
Latest commit History 34 Commits
bh1		bh1
ch1		ch1
edge_resources		edge_resources
helper_functions		helper_functions
k_0_first_order		k_0_first_order
k_1_first_order		k_1_first_order
k_2_first_order		k_2_first_order
k_3_first_order		k_3_first_order
midpoint_resources		midpoint_resources
.gitignore		.gitignore
Lshape_new_ele_files.zip		Lshape_new_ele_files.zip
README.md		README.md
meshgeometry.m		meshgeometry.m
nd1.m		nd1.m
new_ele1.mat		new_ele1.mat
new_ele2.mat		new_ele2.mat
new_ele3.mat		new_ele3.mat
new_ele4.mat		new_ele4.mat
new_ele5.mat		new_ele5.mat
new_ele6.mat		new_ele6.mat
new_ele7.mat		new_ele7.mat
new_ele7_Lshape.mat		new_ele7_Lshape.mat
new_ele8.mat		new_ele8.mat
new_ele9.mat		new_ele9.mat
rt1.m		rt1.m
triquad.m		triquad.m

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

Table of Contents

Introduction

Equations

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

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

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

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

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James Madison University Honors Capstone Project

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

Table of Contents

Introduction

Equations

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

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

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

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

Definitions

Requirements

James Madison University Honors Capstone Project

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