Catálogo de publicaciones - libros

In this paper we recall the construction of the dual finite element complex introduced in [11] and we investigate some applications. More precisely, we propose and analyze fully compatible discretizations for the magnetostatics and the Darcy flow equations in two dimensions, and we introduce an optimal matching condition for domain decomposition methods for Maxwell equations in three dimensions.

The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of the unknowns. The first is the classical pressure instability that occurs when the solid is incompressible, whereas the second is the lack of stability in the stresses. To overcome these instabilities, there are two options. The first is to use different interpolation for all the unknowns satisfying two inf-sup conditions. Whereas there are several displacement-pressure interpolations that render the pressure stable, less possibilities are known for the stress interpolation. The second option is to use a stabilized finite element formulation instead of the plain Galerkin approach. If this formulation is properly designed, it is possible to use equal interpolation for all the unknowns. The purpose of this paper is precisely to present one of such formulations. In particular, it is based on the decomposition of the unknowns into their finite element component and a subscale, that will be approximated and whose goal is to yield a stable formulation. A singular feature of the method to be presented is that the subscales will be considered orthogonal to the finite element space. We describe in detail the original formulation and a simplified variant and present the results of their numerical analysis.

We investigate adaptive wavelet methods which are in the sense that a of the solution of a linear elliptic PDE is computed up to arbitrary accuracy at possibly low computational cost measured in terms of degrees of freedom. In particular, we propose a scheme that can be shown to exhibit convergence to the target value without insisting on energy norm convergence of the primal solution. The theoretical findings are complemented by first numerical experiments.

We first review our recent results concerning optimal algorithms for the solution of bound and/or equality constrained quadratic programming problems. The unique feature of these algorithms is the rate of convergence in terms of bounds on the spectrum of the Hessian of the cost function. Then we combine these estimates with some results on the FETI method (FETI-DP, FETI and Total FETI) to get the convergence bounds that guarantee the scalability of the algorithms. i.e. asymptotically linear complexity and the time of solution inverse proportional to the number of processors. The results are confirmed by numerical experiments.

This work presents a unified analysis of Discontinuous Galerkin methods to approximate Friedrichs’ systems. A general set of boundary conditions is identified to guarantee existence and uniqueness of solutions to these systems. A formulation enforcing the boundary conditions weakly is proposed. This formulation is the starting point for the construction of Discontinuous Galerkin methods formulated in terms of boundary operators and of interface operators that mildly penalize interface jumps. A general convergence analysis is presented. The setting is subsequently specialized to two-field Friedrichs’ systems endowed with a particular 2×2 structure in which some of the unknowns can be eliminated to yield a system of second-order elliptic-like PDE’s for the remaining unknowns. A general Discontinuous Galerkin method where the above elimination can be performed in each mesh cell is proposed and analyzed. Finally, details are given for four examples, namely advection-reaction equations, advection-diffusion-reaction equations, the linear elasticity equations in the mixed stress-pressure-displacement form, and the Maxwell equations in the so-called elliptic regime.

The last few years have witnessed substantive developments in the computation of highly oscillatory integrals in one or more dimensions. The availability of new asymptotic expansions and a Stokes-type theorem allow for a comprehensive analysis of a number of old (although enhanced) and new quadrature techniques: the asymptotic, Filon-type and Levin-type methods. All these methods share the surprising property that their accuracy increases with growing oscillation.

We use the linear sampling method to determine the shape and surface conductivity of a partially coated dielectric from a knowledge of the far field pattern of the scattered electromagnetic wave at fixed frequency. A mathematical justification of the method is provided for the full 3D vector case based on the use of a complete family of solutions. Numerical examples are given.

The paper is concerned with functional approach to the a posteriori error control for approximate solutions of differential equations. Functional a posteriori estimates are derived by purely functional methods using the analysis of variational problems or integral identities. They are intended to give computable minorants and majorants for various measures of the difference between exact solutions and their conforming approximations. Functional estimates contain no mesh dependent constants and provide guaranteed lower and upper bounds of errors. In this paper, the major attention is paid on a posteriori estimates in terms of local norms or locally based linear functionals. It is shown that for linear elliptic and parabolic problems functional estimates in global (energy) norms imply a posteriori estimates in terms of local quantities.

In this paper we consider a problem of parabolic optimal design in 2D for the heat equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. The discrete optimal design problem admits at least one solution.