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

Introduction

This repository contains efficient lowest order Fourier finite element methods to approximate the solution of Hodge Laplacian problems on axisymmetric domains.

In [1], a new family of Fourier finite element spaces was constructed by using the lowest order finite element methods. These spaces were used to discretize Hodge Laplacian problems in [2]. This repository contains open-source programs for each of the four problems described as well as some additional programs for the spaces used to construct the new family of spaces.

This repository is written in conjuction with this repository, which contains the higher 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

Other Equations

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

References

[1] Minah Oh, de Rham complexes arising from Fourier finite element methods in axisymmetric domains, Computers & Mathematics with Applications, Volume 70, Issue 8, 2015
[2] The Hodge Laplacian on axisymmetric domains and its discretization, IMA Journal of Numerical Analysis, 2020

James Madison University Honors Capstone Project

Author: Nicole Stock
Faculty Research Advisor: Dr. Minah Oh

Name		Name	Last commit message	Last commit date
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data		data
edge_resources		edge_resources
k_0		k_0
k_1		k_1
k_2		k_2
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midpoint_resources		midpoint_resources
modified_weighted_nedelec_pk		modified_weighted_nedelec_pk
nedelec		nedelec
raviart_thomas		raviart_thomas
.gitignore		.gitignore
README.md		README.md
display_errors.m		display_errors.m
errors_no_exact_weighted_HL_p2.m		errors_no_exact_weighted_HL_p2.m
find_edge_connectivity.m		find_edge_connectivity.m
find_midpoints.m		find_midpoints.m
get_edge_connectivity.m		get_edge_connectivity.m
get_midpoints.m		get_midpoints.m
prolongation_matrix.m		prolongation_matrix.m
triquad.m		triquad.m

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Lowest 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

Other Equations

Weighted Lowest Order Nedelec Space

Weighted Fourier Lowest Order Nedelec and P1 Space (Bh)

Lowest Order Raviart Thomas Space

Weighted Fourier Lowest Order Raviart Thomas Space (Ch)

Definitions

Requirements

References

James Madison University Honors Capstone Project

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