### Browsing by Author "Barrenechea, Gabriel R."

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- ItemAnalysis of algebraic flux correction schemes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Barrenechea, Gabriel R.; John, Volker; Knobloch, PetrA 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.
- ItemA 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, PetrAn 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.
- ItemA pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Ahmed, Naveed; Barrenechea, Gabriel R.; Burman, Erik; Guzmán, Johnny; Linke, Alexander; Merdon, ChristianDiscretization of Navier--Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressureindependent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an O(hk+1/2) error estimate in the L2-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization.
- ItemSome 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, PetrAlgebraic 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.
- ItemA 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, RichardRecent 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.
- ItemA unified analysis of algebraic flux correction schemes for convection–diffusion equations(Berlin ; Heidelberg : Springer, 2018) Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr; Rankin, RichardRecent 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 schemes. Then, specific versions of the method, that is, different definitions for the flux limiters, are reviewed and their main results stated. Numerical studies compare the different versions of the scheme.