08:30
25 mins
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Recent advances for exactly incompressible elements
Ridgway Scott
Abstract: Finite element approximations satisfying exact incompressibility conditions are
now recognized as essential for certain flow simulations. We discuss several
enhancements for algorithms based on what is known as the Scott-Vogelius
method. One is known as the unified Stokes algorithm (USA) and projects
the discontinuous pressure arising in Scott-Vogelius onto a continuous
pressure space. Another enhancement relates to multi-grid solvers. It
involves new smoothers that preserve the incompressibility condition and
insure optimal convergence. A third relates to new results concerning
the validity of the inf-sup condition for lower-degree approximations.
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08:55
25 mins
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A space-time HDG method for incompressible flows
Tamas Horvath, Keegan Kirk, Sander Rhebergen
Abstract: In this talk we present a space-time hybridizable discontinuous Galerkin (HDG) finite element method for the Navier-Stokes equations on time-dependent domains [1]. For this we discretize the space-time formulation of the Navier-Stokes equations on a (d+1)-dimensional space-time domain. The space-time domain is partitioned into (d+1)-simplices on which the HDG function spaces are defined. These function spaces therefore take into account the domain deformation. We can then apply a divergence-free (d+1)-dimensional HDG method to the space-time formulation of the Navier-Stokes equations. The result is a discretization of the Navier-Stokes equations in which the velocity approximation is divergence-free and H(div)-conforming, even on time-dependent (deforming) domains.
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09:20
25 mins
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Finite element discretizations with exactly tangential vector fields for incompressible flows on surfaces
Christoph Lehrenfeld, Philip L. Lederer, Joachim Schöberl
Abstract: We consider a surface Navier-Stokes model for a Newtonian surface fluid on a two-dimensional surface embedded in three dimensions.
For the discretization of the velocity field we use an H(div)-conforming finite element space. This is achieved by mapping finite element functions from the two-dimensional reference element by a straight-forward generalization of the well-known Piola transformation.
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