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Higher Order Finite Element Methods to Approximate Hodge Laplacian Problems on an Axisymmetric Domain

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

Table of Contents

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 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

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

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

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

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Higher Order Finite Element Methods to Approximate Hodge Laplacian Problems on an Axisymmetric Domain

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