Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations

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Date
2016
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Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Abstract

Recently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new pressure-robust Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously. In this contribution, pressure-robustness is extended to the time-dependent Navier-Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the discrete Helmholtz projector of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed.

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Keywords
Incompressible Navier–Stokes, mixed finite element methods, a-priori error estimates, pressure-robustness, Helmholtz projector, irrotational forces
Citation
Citation
Linke, A., & Merdon, C. (2016). Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations (Version publishedVersion, Vol. 2250). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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