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Type: Report × Subject: Mixed finite elements ×

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Now showing 1 - 4 of 4
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    Stabilized finite element schems for incompressible flow using Scott-Vogelius elements
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Burman, Erik; Linke, Alexander
    We propose a stabilized finite element method based on the Scott-Vogelius element in combination with either a local projection stabilization or an edge oriented stabilization based on a penalization of the gradient jumps over element edges. We prove a discrete inf-sup condition leading to optimal a priori error estimates. The theoretical considerations are illustrated by some numerical examples.
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    Guaranteed energy error estimators for a modified robust Crouzeix-Raviart Stokes element
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Linke, Alexander; Merdon, Christian
    This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix-Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method.
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    On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Linke, Alexander
    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 L2-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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    Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Galvin, Keith J.; Linke, Alexander; Rebholz, Leo G.; Wilson, Nicholas E.
    We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation-free forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergence-free finite elements (such as Scott-Vogelius), or heavy grad-div stabilization of weakly divergence-free elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers.