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On the Role of the Helmholtz Decomposition in Mixed Methods for Incompressible Flows and a New Variational Crime

Please always quote using this URN:urn:nbn:de:0296-matheon-12380
  • According to the Helmholtz decomposition, the irrotational parts of the momentum balance equations of the incompressible Navier-Stokes equations are balanced by the pressure gradient. Unfortunately, nearly all mixed methods for incompressible flows violate this fundamental property, resulting in the well-known numerical instability of poor mass conservation. The origin of this problem is the lack of $L^2$-orthogonality between discretely divergence-free velocities and irrotational vector fields. In order to cure this, a new variational crime using divergence-free velocity reconstructions is proposed. Applying lowest order Raviart-Thomas velocity reconstructions to the nonconforming Crouzeix-Raviart element allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence-free flow solvers. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

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Metadaten
Author:Alexander Linke
URN:urn:nbn:de:0296-matheon-12380
Referee:Ralf Kornhuber
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2013/09/03
Release Date:2013/09/03
Tag:Helmholtz decomposition; Incompressible Navier-Stokes Equations; mixed finite elements; poor mass conservation
Institute:Freie Universität Berlin
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Preprint Number:1031
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