A unified framework for the analysis, numerical approximation and model reduction of linear operator equations, Part I: Well-posedness in space and time

TL;DR

This paper introduces a unified theoretical framework for analyzing, approximating, and reducing linear operator equations across various types of PDEs, establishing a foundation for future numerical and model reduction techniques.

Contribution

It develops a comprehensive approach to well-posed formulations for diverse linear operator equations, including novel space-time variational forms for hyperbolic PDEs.

Findings

Framework unifies treatment of elliptic, parabolic, hyperbolic equations

Extends operators from strong to weak formulations

Lays groundwork for numerical approximation and model reduction

Abstract

We present a unified framework to construct well-posed formulations for large classes of linear operator equations including elliptic, parabolic and hyperbolic partial differential equations. This general approach incorporates known weak variational formulations as well as novel space-time variational forms of the hyperbolic wave equation. The main concept is completion and extension of operators starting from the strong form of the problem. This paper lays the theoretical foundation for a unified approach towards numerical approximation methods and also model reduction of parameterized linear operator equations which will be the subject of the following parts.