New Laplace and Helmholtz solvers

TL;DR

This paper introduces novel rational function-based algorithms for solving Laplace and Helmholtz equations on 2D domains with corners, achieving faster and more accurate results than traditional finite element and integral equation methods.

Contribution

The paper presents new numerical algorithms that challenge existing assumptions in PDE analysis, offering improved efficiency and accuracy for specific boundary value problems.

Findings

Faster solution times compared to standard methods

Higher accuracy in corner domain problems

Potential to reshape numerical PDE analysis

Abstract

New numerical algorithms based on rational functions are introduced that can solve certain Laplace and Helmholtz problems on two-dimensional domains with corners faster and more accurately than the standard methods of finite elements and integral equations. The new algorithms point to a reconsideration of the assumptions underlying existing numerical analysis for partial differential equations.