A least-squares space-time approach for parabolic equations

Michael Hinze; Christian Kahle; Michael Stahl

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arXiv · 2305.03402

A least-squares space-time approach for parabolic equations

Michael Hinze·Christian Kahle·Michael Stahl

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Abstract

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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Michael Hinze, Christian Kahle, Michael Stahl. 2023-05-05. A least-squares space-time approach for parabolic equations. https://arxiv.org/abs/2305.03402

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