On a mixed FEM and a FOSLS with H^-1 loads

2022/10/25 by Thomas Führer, Führer, Thomas · 2 citations
Engineering · #65N12 #65N30 #Advanced Numerical Methods in Computational Mathematics #Composite Structure Analysis and Optimization #Elasticity and Material Modeling #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2210.14063

openalex publication_date 2022/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study variants of the mixed finite element method (mixed FEM) and the first-order system least-squares finite element (FOSLS) for the Poisson problem where we replace the load by a suitable regularization which permits to use H^-1 loads. We prove that any bounded H^-1 projector onto piecewise constants can be used to define the regularization and yields quasi-optimality of the lowest-order mixed FEM resp. FOSLS in weaker norms. Examples for the construction of such projectors are given. One is based on the adjoint of a weighted Clément quasi-interpolator. We prove that this Clément operator has second-order approximation properties. For the modified mixed method we show optimal convergence rates of a postprocessed solution under minimal regularity assumptions -- a result not valid for the lowest-order mixed FEM without regularization. Numerical examples conclude this work.

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• Adaptive least-squares space-time finite element methods for convection-diffusion problems
• A priori error analysis of the proximal Galerkin method

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