An Approximation Theorist's View on Solving Operator Equations

TL;DR

This paper explores how Approximation Theory can address well-posed PDE problems and operator equations, focusing on approximation and stability issues in standard solution algorithms like Finite Elements and Rayleigh-Ritz.

Contribution

It demonstrates the application of Approximation Theory to analyze and improve common PDE solution methods, providing a new perspective for researchers outside the field.

Findings

Approximation Theory effectively addresses stability issues in PDE algorithms.

Standard methods can be viewed as approximation problems.

The approach offers insights into improving solution stability.

Abstract

When an Approximation Theorist looks at well-posed PDE problems or operator equations, and standard solution algorithms like Finite Elements, Rayleigh-Ritz or Trefftz techniques, methods of fundamental or particular solutions and their combinations, they boil down to approximation problems and stability issues. These two can be handled by Approximation Theory, and this paper shows how, with special applications to the aforementioned algorithms. The intention is that the Approximation Theorists viewpoint is helpful for readers who are somewhat away from that subject.