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search.filters.applied.f.access: openAccess × Author: John, Volker × Type: Text × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: error analysis ×

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    Finite element pressure stabilizations for incompressible flow problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) John, Volker; Knobloch, Petr; Wilbrandt, Ulrich
    Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis.
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    Error analysis of a SUPG-stabilized POD-ROM method for convection-diffusion-reaction equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) John, Volker; Moreau, Baptiste; Novo, Julia
    A reduced order model (ROM) method based on proper orthogonal decomposition (POD) is analyzed for convection-diffusion-reaction equations. The streamline-upwind Petrov--Galerkin (SUPG) stabilization is used in the practically interesting case of dominant convection, both for the full order method (FOM) and the ROM simulations. The asymptotic choice of the stabilization parameter for the SUPG-ROM is done as proposed in the literature. This paper presents a finite element convergence analysis of the SUPG-ROM method for errors in different norms. The constants in the error bounds are uniform with respect to small diffusion coefficients. Numerical studies illustrate the performance of the SUPG-ROM method.
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    Error analysis of the SUPG finite element disretization of evolutionary convection-diffusion-reaction equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) John, Volker; Novo, Julia
    Conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the SUPG finite element stabilization of time-dependent convection-diffusion-reaction equations that is combined with the backward Euler method. Standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step. The stabilization vanishes in the time-continuous limit. However, based on numerical experiences, this seems not to be the correct behavior. For this reason, the time-continuous case is analyzed under certain conditions on the coefficients of the equation and the finite element method. An error estimate with the standard order of convergence is derived for stabilization parameters of the same form that is optimal for the steady-state problem. Numerical studies support the analytical results.
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    A unified analysis of Algebraic Flux Correction schemes for convection-diffusion equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr; Rankin, Richard
    Recent results on the numerical analysis of Algebraic Flux Correction (AFC) finite element schemes for scalar convection-diffusion equations are reviewed and presented in a unified way. A general form of the method is presented using a link between AFC schemes and nonlinear edge-based diffusion scheme. Then, specific versions of the method, this is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme.