On the necessity of the inf-sup condition for a mixed finite element formulation

2022/06/14 by Fleurianne Bertrand, Daniele Boffi, Bertrand, Fleurianne +1
Engineering · Mathematics · #65N12 #65N30 #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2206.06968

openalex publication_date 2022/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We study a non standard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in H(div) for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.

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