An Approximation Theorist's View on Solving Operator Equations
For researchers in numerical PDEs, this provides a unifying theoretical framework, but it is primarily a conceptual exposition without new numerical results.
This paper reframes well-posed PDE problems and standard solution algorithms as approximation problems and stability issues, demonstrating how Approximation Theory can handle them. It applies this perspective to Finite Elements, Rayleigh-Ritz, Trefftz, and method of fundamental solutions.
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.