A Generic MATLAB Toolbox to Approximate PDEs Using Computational Geometry

TL;DR

This paper presents a versatile MATLAB toolbox that leverages computational geometry to efficiently approximate PDEs, automating sparsity pattern identification and integrating advanced quadrature methods for improved performance.

Contribution

It introduces a general framework and software that automatically identifies sparsity patterns and incorporates efficient quadrature methods for PDE approximation.

Findings

Effective approximation of PDEs demonstrated on standard examples

Automated sparsity pattern detection simplifies implementation

Quadrature methods outperform traditional tensor product approaches

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.