Blanca Ayuso de Dios

DISCONTINUOUS GALERKIN METHODS FOR THE MULTI-DIMENSIONAL VLASOV–POISSON PROBLEM

We introduce and analyze two new semi-discrete numerical methods for the multi-dimensional Vlasov–Poisson system. The schemes are constructed by combining a discontinuous Galerkin approximation to the Vlasov equation together with a mixed finite element method for the Poisson problem. We show optimal error estimates in the case of smooth compactly supported initial data. We propose a scheme that preserves the total energy of the system.

MULTILEVEL PRECONDITIONERS FOR DISCONTINUOUS GALERKIN APPROXIMATIONS OF ELLIPTIC PROBLEMS WITH JUMP COEFFICIENTS

In this article we develop and analyze two-level and multi-level methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coecients (exhibiting large jumps across interfaces in the domain). These methods are based on a decomposition of the DG nite element space that inherently hinges on the diusion coecient of the problem. Our analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes, and we establish both robustness with respect to the jump in the coecient and near-optimality with respect to the mesh size. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods.

A Simple Preconditioner for a Discontinuous Galerkin Method for the Stokes Problem

In this paper we construct Discontinuous Galerkin approximations of the Stokes problem where the velocity field is

Stability Analysis of Discontinuous Galerkin Approximations to the Elastodynamics Problem

H(\mathrm{div},\Omega )

Multigrid Preconditioner for Nonconforming Discretization of Elliptic Problems with Jump Coefficients

H(div,Ω)-conforming. This implies that the velocity solution is divergence-free in the whole domain. This property can be exploited to design a simple and effective preconditioner for the final linear system.

Space Decompositions and Solvers for Discontinuous Galerkin Methods

Abstract We propose and analyze several two-level non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) discretization of a second order boundary value problem. We show that the preconditioners are scalable and that the condition number of the resulting preconditioned linear systems of equations is independent of the penalty parameter and is of order H/h, where H and h represent the mesh sizes of the coarse and fine partitions, respectively. Numerical experiments that illustrate the performance of the proposed two-level Schwarz methods are also presented.

The nonconforming virtual element method

We consider semi-discrete discontinuous Galerkin approximations of both displacement and displacement-stress formulations of the elastodynamics problem. We prove the stability analysis in the natural energy norm and derive optimal a-priori error estimates. For the displacement-stress formulation, schemes preserving the total energy of the system are introduced and discussed. We verify our theoretical estimates on two and three dimensions test problems.

A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations

We consider an exponentially fitted discontinuous Galerkin method for advection dominated problems and propose a block solver for the resulting linear systems. In the case of strong advection the solver is robust with respect to the advection direction and the number of unknowns.