Michael Stahl | Searcharxiv

arXiv2026

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

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.

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arXiv2025

A least-squares space-time approach for parabolic equations

We propose a least squares formulation for abstract parabolic equations in the natural $L^2(0,T;V^\star)\times H$ norm which only relies on natural regularity assumptions on the data of the problem. The resulting bilinear form then is symmetric, coercive and continuous. We provide two space-time Galerkin frameworks for the numerical approximation. The first one uses a conformal discretization of the underlying bilinear system and relies on the fact that the $V^*-$norm of basis functions can be evaluated exactly. The second approach is nonconforming an replaces the evaluation of the $V^*-$norm by a discrete pendant. We prove convergence for both approaches and illustrate our analytical findings by selected numerical experiments.

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