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A Generic MATLAB Toolbox to Approximate PDEs Using Computational Geometry

Kiefer Green, Harbir Antil

Abstract

This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from computational geometry. They may enable experimentation with novel mesh generation techniques and could simplify the implementation of methods such as multigrid. We also implement quadrature methods following the work of Grundmann and Moller. These methods have been overlooked in the past but are more efficient than traditional tensor product methods. The proposed framework is applied to several standard examples.

A Generic MATLAB Toolbox to Approximate PDEs Using Computational Geometry

Abstract

This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from computational geometry. They may enable experimentation with novel mesh generation techniques and could simplify the implementation of methods such as multigrid. We also implement quadrature methods following the work of Grundmann and Moller. These methods have been overlooked in the past but are more efficient than traditional tensor product methods. The proposed framework is applied to several standard examples.

Paper Structure

This paper contains 10 sections, 5 theorems, 39 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. General Remarks
  3. Finite Element Spaces
  4. Geometry
  5. Quadrature
  6. Numerical Experiments
  7. Poisson Problem
  8. Convection-Diffusion Problem
  9. Non-Linear-Diffusion Problem
  10. Stokes Problem

Key Result

Theorem 4.1

If $a(u,v)=\int_\Omega f(u,x)g(v,x)dx$ where $f$ and $g$ are Support-Preserving. Then, where $\mathop{\mathrm{supp}}\nolimits{u}\subset S_u$ and $\mathop{\mathrm{supp}}\nolimits{v}\subset S_v$.

Figures (6)

  • Figure 1: A conforming triangulation of the floor plan of the fourth floor of Exploratory Hall at George Mason University, Fairfax, Virginia.
  • Figure 2: The left panel shows that if there is a separating axis, the dashed gray line then convex sets are disjoint and the right panel shows that if two convex sets are disjoint then the axis connecting their closest points separates.
  • Figure 3: Left panel shows the approximate solution of Example \ref{['ex1']} using a uniform triangular mesh with maximum side length $h= \frac{1}{8}$. The right panel shows $L_2$ error in approximating Example \ref{['ex1']} using uniform triangular meshes, varying the maximal side length $h$. The expected quadratic rate of convergence is observed.
  • Figure 4: This figure shows the approximate solution of the convection-diffusion system \ref{['ex2']} with a unit impulse load located at $(x,y)=(\frac{1}{2},1)$. This is approximated with a uniform triangular mesh whose maximal side length is $h=\frac{1}{32}$.
  • Figure 5: The left panel shows the approximate solution of Example \ref{['ex3']} using uniformly spaced nodes with spacing $h= \frac{1}{256}$ and the right panel shows the quadratic rate of convergence.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Definition 2.1: Support Preserving
  • Definition 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 5.1
  • Theorem 5.2: Bramble-Hilbert BrennerScott
  • Theorem 5.3
  • proof