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Author: Linke, Alexander × Subject: Finite-Elemente-Methode ×

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    On a reduced sparsity stabilization of grad-div type for incompressible flow problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Linke, Alexander; Rebholz, Leo
    We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operators matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar.
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    Efficient linear solvers for incompressible flow simulations using Scott-Vogelius finite elements
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Cousins, Benjamin; Le Borne, Sabine; Linke, Alexander; Rebholz, Leo G.; Wang, Zhen
    Recent research has shown that in some practically relevant situations like multi-physics flows [11] divergence-free mixed finite elements may have a significantly smaller discretization error than standard nondivergence-free mixed finite elements. In order to judge the overall performance of divergence-free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in ((Pk)d; Pdisc k-1 )) Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott-Vogelius elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements like Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes, and discuss extensions. For iterative methods, we test augmented Lagrangian and H