Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929533413752832 |
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| author | Hajduk, Hennes Kuzmin, Dmitri Lube, Gert Öffner, Philipp |
| author_facet | Hajduk, Hennes Kuzmin, Dmitri Lube, Gert Öffner, Philipp |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06174 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations Hajduk, Hennes Kuzmin, Dmitri Lube, Gert Öffner, Philipp Numerical Analysis 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. |
| title | Locally energy-stable finite element schemes for incompressible flow problems: Design and analysis for equal-order interpolations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2410.06174 |