Blanca Ayuso

Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems

We apply the weighted-residual approach recently introduced in [F. Brezzi et al., Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 3293-3310] to derive discontinuous Galerkin formulations for advection-diffusion-reaction problems. We devise the basic ingredients to ensure stability and optimal error estimates in suitable norms, and propose two new methods.

The Postprocessed Mixed Finite-Element Method for the Navier--Stokes Equations

A postprocessing technique for mixed finite-element methods for the incompressible Navier--Stokes equations is studied. The technique was earlier developed for spectral and standard finite-element methods for dissipative partial differential equations. The postprocessing amounts to solving a Stokes problem on a finer grid (or higher-order space) once the time integration on the coarser mesh is completed. The analysis presented here shows that this technique increases the convergence rate of both the velocity and the pressure approximations. Numerical experiments are presented that confirm both this increase in the convergence rate and the corresponding improvement in computational efficiency.

REGULARITY CONSTANTS OF THE STOKES PROBLEM. APPLICATION TO FINITE-ELEMENT METHODS ON CURVED DOMAINS

For bounded smooth domains, we study how the solution of the Stokes problem is bounded in terms of the data when the domain changes. We show that standard bounds for a fixed domain hold with the same constant for wide classes of domains. This is done by first reviewing the original results of Cattabriga and then, in terms of geometric properties of the domains, by specifying when to apply Cattabrigas intermediate results. We do the same with standard strong extension and trace theorems. We apply these results to an elegant technique of analysis due to Wahlbin to overcome the disparity between a curved domain and the domains where finite-element computations are carried out in practice without resorting to numerical quadrature.

Class of Preconditioners for Discontinuous Galerkin Approximations of Elliptic Problems

Domain decomposition (DD) methods provide powerful preconditioners for the iterative solution of the large algebraic linear systems of equations that arise in finite element approximations of partial differential equations. Many DD algorithms can conveniently be described and analyzed as Schwarz methods, and, if on the one hand a general theoretical framework has been previously developed for classical conforming discretizations (see, e.g., [7]), on the other hand, only few results can be found for discontinuous Galerkin (DG) approximations (see, e.g., [6, 4, 2, 1]). Based on discontinuous finite element spaces, DG methods have become increasing popular thanks to their great flexibility for providing discretizations on matching and non-matching grids and their high degree of locality. In this paper we present and analyze, in the unified framework based on the flux formulation proposed in [3], a class of Schwarz preconditioners for DG approximations of second order elliptic problems. Schwarz methods for a wider class of DG discretizations are studied in [2, 1]. The issue of preconditioning non-symmetric DG approximations is also discussed. Numerical experiments to asses the performance of the proposed preconditioners and validate our convergence results are presented.

Numerical methods for modelling leaching of pollutants in soils

Linear equilibrium and non-equilibrium models for leaching of solutes in soils give rise to unsteady linear convection-diffusion-reaction problems. We present several numerical schemes to approximate the solution of this kind of problems based on Stabilized Finite Element Methods, including the recent Link-Cutting Bubbles strategy adapted to deal with unsteady problems, which gives the best numerical results.

A Simple Uniformly Convergent Iterative Method for the Non-symmetric Incomplete Interior Penalty Discontinuous Galerkin Discretization

We introduce a uniformly convergent iterative method for the systems arising from non-symmetric IIPG linear approximations of second order elliptic problems. The method can be viewed as a block Gaus–Seidel method in which the blocks correspond to restrictions of the IIPG method to suitably constructed subspaces. Numerical tests are included, showing the uniform convergence of the iterative method in an energy norm.

Schwarz Domain Decomposition Preconditioners for Interior Penalty Approximations of Elliptic Problems

We present a two-level non-overlapping additive Schwarz method for Discontinuous Galerkin approximations of elliptic problems. In particular, a two level-method for both symmetric and non-symmetric schemes will be considered and some interesting features, which have no analog in the conforming case, will be discussed. Numerical experiments on non-matching grids will be presented.