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Type: report × Author: John, Volker ×

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Now showing 1 - 10 of 43
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    On basic iteration schemes for nonlinear AFC discretizations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Jha, Abhinav; John, Volker
    Algebraic flux correction (AFC) finite element discretizations of steady-state convection-diffusionreaction equations lead to a nonlinear problem. This paper presents first steps of a systematic study of solvers for these problems. Two basic fixed point iterations and a formal Newton method are considered. It turns out that the fixed point iterations behave often quite differently. Using a sparse direct solver for the linear problems, one of them exploits the fact that only one matrix factorization is needed to become very efficient in the case of convergence. For the behavior of the formal Newton method, a clear picture is not yet obtained.
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    On reducing spurious oscillations in discontinuous Galerkin (DG) methods for steady-state convection-diffusion-reaction equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Frerichs, Derk; John, Volker
    A standard discontinuous Galerkin (DG) finite element method for discretizing steady-state convection-diffusion-reaction equations is known to be stable and to compute sharp layers in the convection-dominated regime, but also to show large spurious oscillations. This paper studies post-processing methods for reducing the spurious oscillations, which replace the DG solution in a vicinity of layers by a constant or linear approximation. Three methods from the literature are considered and several generalizations and modifications are proposed. Numerical studies with the post-processing methods are performed at two-dimensional examples.
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    On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) John, Volker; Novo, Julia
    Finite element and finite difference discretizations for evolutionary convection-diffusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank--Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge--Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods.
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    Analysis of a full space-time discretization of the Navier-Stokes equations by a local projection stabilization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Ahmed, Naveed; Rebollo, Tomás Chacón; John, Volker; Rubino, Samuele
    A finite element error analysis of a local projection stabilization (LPS) method for the time-dependent Navier-Stokes equations is presented. The focus is on the highorder term-by-term stabilization method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard LPS methods is replaced by an interpolation-stabilized structure. The main contribution is on proving, theoretically and numerically, the optimal convergence order of the arising fully discrete scheme. In addition, the asymptotic energy balance is obtained for slightly smooth flows. Numerical studies support the analytical results and illustrate the potential of the method for the simulation of turbulent ows. Smooth unsteady flows are simulated with optimal order of accuracy.
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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.
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    Analysis of algebraic flux correction schemes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr
    A family of algebraic flux correction schemes for linear boundary value problems in any space dimension is studied. These methods main feature is that they limit the fluxes along each one of the edges of the triangulation, and we suppose that the limiters used are symmetric. For an abstract problem, the existence of a solution, existence and uniqueness of the solution of a linearized problem, and an a priori error estimate, are proved under rather general assumptions on the limiters. For a particular (but standard in practice) choice of the limiters, it is shown that a local discrete maximum principle holds. The theory developed for the abstract problem is applied to convection-diffusion-reaction equations, where in particular an error estimate is derived. Numerical studies show its sharpness.
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    Analysis of the PSPG stabilization for the continuous-in-time discretization of the evolutionary stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) John, Volker; Novo, Julia
    Optimal error estimates for the pressure stabilized Petrov-Galerkin (PSPG) method for the continuous-in-time discretization of the evolutionary Stokes equations are proved in the case of regular solutions. The main result is applicable to higher order finite elements. The error bounds for the pressure depend on the error of the pressure at the initial time. An approach is suggested for choosing the discrete initial velocity in such a way that this error is bounded. The "instability of the discrete pressure for small time steps", which is reported in the literature, is discussed on the basis of the analytical results. Numerical studies confirm the theoretical results, showing in particular that this instability does not occur for the proposed initial condition.
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    An adaptive SUPG method for evolutionary convection-diffusion equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) de Frutos, Javier; García-Archilla, Bosco; John, Volker; Novo, Julia
    An adaptive algorithm for the numerical simulation of time-dependent convectiondiffusion-reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convectiondiffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from [18] will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in [6]
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    On iterative subdomain methods for the Stokes-Darcy problem
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Caiazzo, Alfonso; John, Volker; Wilbrandt, Ulrich
    Iterative subdomain methods for the StokesDarcy problem that use Robin boundary conditions on the interface are reviewed. Their common underlying structure and their main differences are identified. In particular, it is clarified that there are different updating strategies for the interface conditions. For small values of fluid viscosity and hydraulic permeability, which are relevant in applications from geosciences, it is shown in numerical studies that only one of these updating strategies leads to an efficient numerical method, if this strategy is used in combination with appropriate parameters in the Robin boundary conditions. In particular, it is observed that the values of appropriate parameters are larger than those proposed so far. Not only the size but also the ratio of appropriate Robin parameters depends on the coefficients of the problem.
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    Projection methods for incompressible flow problems with WENO finite difference schemes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Frutos, Javier de; John, Volker; Novo, Julia
    Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary convectiondiffusion equations [?]. This paper explores the applicability of these schemes for the simulation of incompressible flows. To this end, WENO schemes are used in several nonincremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious.