A reduced basis method for parabolic PDEs based on a space-time least squares formulation

TL;DR

This paper introduces a POD-greedy reduced basis method for parabolic PDEs using a space-time least squares formulation, enabling efficient parameter-dependent problem solutions with error certification.

Contribution

It extends the least squares space-time approach to parameter-dependent parabolic PDEs and develops a reduced basis method with offline-online decomposition and error bounds.

Findings

Method achieves efficient reduced-order solutions for parameterized parabolic PDEs.

Provides rigorous error bounds and certification for the reduced basis solutions.

Demonstrates effectiveness through numerical examples.

Abstract

In this work, we present a POD-greedy reduced basis method for parabolic partial differential equations (PDEs), based on the least squares space-time formulation proposed in [Hinze, Kahle, Stahl, A least-squares space-time approach for parabolic equations, 2023, arXiv:2305.03402] that assumes only minimal regularity. We extend this approach to the parameter-dependent case. The corresponding variational formulation then is based on a parameter-dependent, symmetric, uniformly coercive, and continuous bilinear form. We apply the reduced basis method to this formulation, following the well-developed techniques for parameterized coercive problems, as seen e.g. in reduced basis methods for parameterized elliptic PDEs. We present an offline-online decomposition and provide certification with absolute and relative error bounds. The performance of the method is demonstrated using selected numerical examples.