A least-squares space-time approach for parabolic equations

2023/05/05 by Michael Hinze, Hinze, Michael, Christian Kahle +2
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Numerical methods for differential equations #Computational Fluid Dynamics and Aerodynamics

paper · pdf · doi:10.48550/arxiv.2305.03402

Abstract

We propose a least squares formulation for abstract parabolic equations in the natural L²(0,T;V^⋆)× 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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