A computational tool for comparing all linear PDE solvers -- Optimal methods are meshless

TL;DR

This paper introduces a computational framework to compare all linear PDE solvers using a unified norm-based criterion, revealing that the optimal method is a meshless, symmetric collocation approach with superior performance.

Contribution

It develops a universal comparison method for linear PDE solvers and identifies a unique, optimal meshless technique based on Sobolev space norms.

Findings

A norm-based criterion enables fair comparison of linear PDE methods.

The optimal method is meshless and coincides with symmetric collocation using the Hilbert space kernel.

This optimal method outperforms finite-element, finite-difference, and other meshless techniques.

Abstract

The paper starts out with a computational technique that allows to compare all linear methods for PDE solving that use the same input data. This is done by writing them as linear recovery formulas for solution values as linear combinations of the input data. Calculating the norm of these reproduction formulas on a fixed Sobolev space will then serve as a quality criterion that allows a fair comparison of all linear methods with the same inputs, including finite-element, finite-difference and meshless local Petrov-Galerkin techniques. A number of illustrative examples will be provided. As a byproduct, it turns out that a unique error--optimal method exists. It necessarily outperforms any other competing technique using the same data, e.g. those just mentioned, and it is necessarily meshless, if solutions are written "entirely in terms of nodes" (Belytschko et. al. 1996). On closer inspection, it turns out that it coincides with {\em symmetric meshless collocation} carried out with the kernel of the Hilbert space used for error evaluation, e.g. with the kernel of the Sobolev space used. This technique is around since at least 1998, but its optimality properties went unnoticed, so far.