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Author: John, Volker × End date: 2019 × Start date: 2010 × Type: Text × Subject: Finite element method × WGL Institute: WIAS ×

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    A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr
    An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.
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    Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in 1D
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr
    Algebraic flux correction schemes are nonlinear discretizations of convection dominated problems. In this work, a scheme from this class is studied for a steady-state convection-diffusion equation in one dimension. It is proved that this scheme satisfies the discrete maximum principle. Also, as it is a nonlinear scheme, the solvability of the linear subproblems arising in a Picard iteration is studied, where positive and negative results are proved. Furthermore, the non-existence of solutions for the nonlinear scheme is proved by means of counterexamples. Therefore, a modification of the method, which ensures the existence of a solution, is proposed. A weak version of the discrete maximum principle is proved for this modified method.