A high-order accurate discretization scheme for variable coefficient elliptic PDEs in the plane with smooth solutions

TL;DR

This paper introduces a high-order discretization scheme for variable coefficient elliptic PDEs in the plane, leveraging Gaussian quadratures to efficiently solve smooth scattering problems with linear systems suitable for advanced direct solvers.

Contribution

The paper presents a novel high-order discretization method tailored for variable coefficient elliptic PDEs with smooth solutions, optimized for efficient direct solver compatibility.

Findings

Effective discretization scheme for elliptic PDEs with smooth solutions

Compatibility with nested dissection and accelerated solvers

Achieves O(N) complexity in solving linear systems

Abstract

A discretization scheme for variable coefficient elliptic PDEs in the plane is presented. The scheme is based on high-order Gaussian quadratures and is designed for problems with smooth solutions, such as scattering problems involving soft scatterers. The resulting system of linear equations is very well suited to efficient direct solvers such as nested dissection and the more recently proposed accelerated nested dissection schemes with O(N) complexity.