Catálogo de publicaciones - libros

We will address the problem of finding the minimal necessary stabilization for a class of Discontinuous Galerkin (DG) methods in mixed form. In particular, we will present a new stabilized formulation of the Bassi-Rebay method (see [2] for the original unstable method) and a new formulation of the Local Discontinuous Galerkin (LDG) method (see [5] for the original LDG method).

We develop an -version discontinuous Galerkin method for a nonlinear biharmonic equation corresponding to the two-dimensional incompressible Navier-Stokes equations in the stream-function formulation. We linearize the equation and then we solve the resulting linear problem using a combination of the nonsymmetric discontinuous Galerkin finite element method for the biharmonic part of the equation, and a discontinuous Galerkin finite element method with a jumppenalty term for the hyperbolic part of the equation. Numerical experiments are presented to demonstrate the accuracy of the method for a wide range of Reynolds numbers.

The paper deals with a combination of the Nitsche-mortaring with the Fourier- finite-element method. The approach is applied to the Dirichlet problem of the Poisson equation in three-dimensional axisymmetric domains with nonaxisymmetric data. The approximating Fourier method yields a splitting of the 3D-problem into 2D-problems on the meridian plane treated by the Nitsche- finite-element method (as a mortar method). Some important properties of the approximation scheme as well as error estimates in some -like norm as well as in the -norm are derived.

We study the efficient solution of the linear system arising from the discretization by the mortar method of mathematical models in electrocardiology. We focus on the bidomain extracellular potential problem and on the class of substructuring preconditioners. We verify that the condition number of the preconditioned matrix only grows polylogarithmically with the number of degrees of freedom as predicted by the theory and validated by numerical tests. Moreover, we discuss the role of the conductivity tensors in building the preconditioner.

We apply the continuous interior penalty method to the three fields Stokes problem. We prove an inf-sup condition for the proposed method leading to optimal a priori error estimates for smooth exact solutions. Moreover we propose an iterative algorithm for the separate solution of the velocities and the pressures on the one hand and the extra-stress on the other. The stability of the iterative algorithm is established.

In this work we analyze a projector of non-smooth functions on anisotropic quadrilateral meshes. In particular, a stability result and an upper bound for the approximation error is derived. It turns out that the partial derivatives become well combined with the corresponding mesh size parameters.

A continuous interior penalty -finite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced and analyzed. Error estimates are presented for first-order transport equations. The analysis relies on three technical results that are of independent interest: an -inverse trace inequality, a local discontinuous to continuous -interpolation result, and -error estimates for continuous -orthogonal projections.

We present a nonconforming finite element method with face penalty to approximate advection-diffusion-reaction equations. The a priori error analysis leads to (quasi-)optimal error estimates in the mesh-size keeping the Péclet number fixed. The a posteriori error analysis yields residual-type error indicators that are semi-robust in the sense that the lower and upper bounds of the error differ by a factor bounded by the square root of the Péclet number. Finally, to illustrate the theory, numerical results including adaptively generated meshes are presented.

We study edge-oriented FEM stabilizations w.r.t. linear multigrid solvers and data structures with the goal to examine the efficiency of such stabilizations due to the extending matrix stencil which is not supported by standard FEM data structures. A new edge-oriented data structure has been developed to support the additional coupling. So, the local element-wise and edge-wise matrices are easily deduced from the global ones. Accordingly, efficient Vanka smoothers are introduced, namely a full cell-oriented and an edge-oriented Vanka smoother so that it becomes possible to privilege edge-oriented stabilization for CFD simulations.

We propose a space-time adaptive algorithm for two iterative numerical methods for the solution of nonlinear time depended Landau-Lifshitz-Gilbert equation of micromagnetism. The first method is derived from implicit backward Euler time discretisation, the second method is based on midpoint rule. The space discretisation is done by linear finite elements. The resulting nonlinear systems are solved by an iterative fixed-point technique. The performance of the proposed adaptive strategy is demonstrated by numerical experiments.