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arXiv · 2410.06174

Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations

Hennes Hajduk·Dmitri Kuzmin·Gert Lube·Philipp Öffner
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Abstract

We show that finite element discretizations of incompressible flow problems can be designed to ensure preservation/dissipation of kinetic energy not only globally but also locally. In the context of equal-order (piecewise-linear) interpolations, we prove the validity of a semi-discrete energy inequality for a quadrature-based approximation to the nonlinear convective term, which we combine with the Becker--Hansbo pressure stabilization. An analogy with entropy-stable algebraic flux correction schemes for the compressible Euler equations and the shallow water equations yields a weak `bounded variation' estimate from which we deduce the semi-discrete Lax--Wendroff consistency and convergence towards dissipative weak solutions. The results of our numerical experiments for standard test problems confirm that the method under investigation is non-oscillatory and exhibits optimal convergence behavior.

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Hennes Hajduk, Dmitri Kuzmin, Gert Lube, Philipp Öffner. 2024-10-08. Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations. https://arxiv.org/abs/2410.06174

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