Virginia Selgas

A mixed–FEM and BEM coupling for a three-dimensional eddy current problem

We study in this paper the electromagnetic field generated in an infinite cylindrical conductor by an alternating current density. The resulting interface problem (see [1]) between the metal and the dielectric medium is treated by a mixed-FEM and BEM coupling method. We prove that our BEM-FEM formulation is well posed and leads to a convergent Galerkin method with optimal error estimates. Furthermore, we introduce a fully discrete version of our Galerkin scheme based on simple quadrature formulas. We show that, if the parameter of discretization is sufficiently small, the fully discrete method is well posed and the error estimates remain unaltered.

An inverse acoustic waveguide problem in the time domain

We consider the problem of locating an obstacle in a waveguide from time domain measurements of causal waves. More precisely, we assume that we are given the scattered field due to point sources placed on a surface located inside the waveguide away from the obstacle, where the scattered field is measured on the same surface. From this multi-static scattering data we wish to determine the position and shape of an obstacle in the waveguide. To deal with this inverse problem, we adapt and analyze the time domain linear sampling method. This involves proving new time domain estimates for the forward problem, as well as analyzing several time domain operators arising in the inversion scheme. We also implement the inversion algorithm and provide numerical results in two-dimensions using synthetic data.

Analysis of a splitting-differentiation population model leading to cross-diffusion

Starting from the dynamical system model capturing the splitting-differentiation process of populations, we extend this notion to show how the speciation mechanism from a single species leads to the consideration of several well known evolution cross-diffusion partial differential equations.Among the different alternatives for the diffusion terms, we study the model introduced by Busenberg and Travis, for which we prove the existence of solutions in the one-dimensional spatial case. Using a direct parabolic regularization technique, we show that the problem is well posed in the space of bounded variation functions, and demonstrate with a simple example that this is the best regularity expected for solutions.We numerically compare our approach to other alternative regularizations previously introduced in the literature, for the particular case of the contact inhibition problem. Simulation experiments indicate that the numerical scheme arising from the approximation introduced in this article outperforms those of the existent models from the stability point of view.

A Trefftz Discontinuous Galerkin method for time-harmonic waves with a generalized impedance boundary condition

ABSTRACT We show how a Trefftz Discontinuous Galerkin (TDG) method for the displacement form of the Helmholtz equation can be used to approximate problems having a generalized impedance boundary condition (GIBC) involving surface derivatives of the solution. Such boundary conditions arise naturally when modeling scattering from a scatterer with a thin coating. The thin coating can then be approximated by a GIBC. A second place GIBCs arise is as higher order absorbing boundary conditions. This paper also covers both cases. Because the TDG scheme has discontinuous elements, we propose to couple it to a surface discretization of the GIBC using continuous finite elements. We prove convergence of the resulting scheme and demonstrate it with two numerical examples.

Time Dependent Scattering in an Acoustic Waveguide Via Convolution Quadrature and the Dirichlet-to-Neumann Map

We propose to use finite elements and BDF2 time stepping to solve the problem of computing a solution to the time dependent wave equation with a variable sound speed in an infinite sound hard pipe (waveguide). By using the Laplace transform and an appropriate Dirichlet-to-Neumann (DtN) map for the problem, we can prove that this problem can be reduced to a variational problem on a bounded domain that has a unique solution. This solution can be discretized in space using finite elements (projecting into a Fourier space on the two artificial boundaries to allow the rapid calculation of the DtN map). We discretize in time using the Convolution Quadrature (CQ) approach and in particular BDF2 time-stepping. Thanks to CQ we obtain a stable and convergent discretization of the DtN map, and hence of the fully discrete BDF2-finite element scheme without a CFL condition. We illustrate the method with some numerical results.

On a Splitting-Differentiation Process Leading to Cross-Diffusion

We generalize the dynamical system model proposed by Sanchez-Palencia for the splitting-differentiation process of populations to include spatial dependence. This gives rise to a family of cross-diffusion partial differential equations problems, among which we consider the segregation model proposed by Busenberg and Travis. For the one-dimensional case, we make a direct parabolic regularization of the problem to show the existence of solutions in the space of BV functions. Moreover, we introduce a Finite Element discretization of both our parabolic regularization and an alternative regularization previously proposed in the literature. Our numerical results suggest that our approach is more stable in the tricky regions where the solutions exhibit discontinuities.

Transmission eigenvalues for dielectric objects on a perfect conductor

We present a new interior transmission problem arising when a dielectric structure sits on a perfect conducting plane. The problem has mixed boundary conditions. We discuss the forward problem, and then briefly formulate the standard near field linear sampling method (LSM) for the inverse problem of shape identification. Next we show that the new mixed transmission eigenvalue problem can be analyzed by appropriate modifications to the standard theory of transmission eigenvalues. In particular this involves proving appropriate density and compactness results. We end with some numerical evidence which shows that the LSM can be used for this problem even if limited aperture of data is used. In addition we demonstrate that transmission eigenvalues can be determined from near field scattering data.

Linear Sampling and Reciprocity Gap Methods for a Fluid–Solid Interaction Problem in the Near Field

We consider the inverse problem for a fluid–solid interaction problem of locating and determining the shape of the target from measurements of the near field of the fluid pressure at a single frequency. More precisely we assume that the scattered field and its normal derivative are avaible at any receiver on a measurement surface using fields due to source points around the target for a single frequency. To deal with this problem, we have adapted and analyzed the reciprocity gap method as well as the linear sampling method, showing a novel connection among them. These qualitative approaches are validated by using both synthetic and experimental data in [1].

A Coupling of Spectral and Finite Elements for an Acoustic Scattering Problem

The aim of this paper is to introduce a new fully discrete method for approximating a time-harmonic acoustic wave scattered by a bounded inhomogeneity in the plane. The difficulty related to the unboundness of the domain has been tackled in the literature by different strategies. In particular, the approaches based on finite elements incorporate the far-field effects into the model by means of local (differential) or global absorbing boundary conditions prescribed on an artificial boundary Г enclosing the region of inhomogeneity. Most of the local absorbing boundary conditions are imposed on a circle and they are more exact the larger is the radius; this fact may conduce to large domains or big wave numbers provoking numerical difficulties; cf. [1, 2, 5, 6, 7, 11, 12].