# basque center for applied mathematics

## Simulation of Wave Propagation

As the name of the research line suggests it, we are a group interested in using the computer as a tool to solve mathematical models arising from various real-life problems. In order to obtain reliable algorithms, and therefore valid and trustworthy simulations, a strong mathematical framework is of paramount importance. In this context, we devote a large part of our work to the mathematical analysis of modern numerical methods (finite element method (FEM) and finite difference method (FDM) among others) to be applied to both stationary and time-dependent problems. On the one hand, we are concerned with classical (but still very important) issues related to these methods, as they provide an a priori knowledge about the behavior of the proposed algorithms. More specifically we are addressing problems like convergence, superconvergence, mesh generation, and preservation of qualitative properties of solutions, with emphasize on high-dimensional problems and interconnections with discrete and computational geometry. On the other hand, we are interested in adopting some recent trends in numerical analysis, such as reliable a posteriori error estimates and efficient mesh adaptivity procedures, to reduce the computational costs of the simulations. By combining these two theoretical aspects when designing the new algorithms to be implemented, we can hope to obtain robust and efficient numerical methods to simulate a wide range of applications.

The first application area in our group is the study of scattering problems in bounded domains. The major difficulty encountered when solving numerically Helmholtz problems, that describe time-harmonic wave propagation, is that standard FEM is not suited for mid- and high-frequency regime because of the quasi-optimality constant which grows with the wavenumber. Increasing the number of elements in the mesh and/or the order of the element in order to reach an acceptable level of accuracy leads to a prohibitive computational cost for high wavenumbers. Given that, we are particularly interested in a class of plane wave-based discontinuous Galerkin methods for solving efficiently Helmholtz problems. As it has already been shown, these methods have the potential of reducing considerably the size of the systems to be solved in standard FEM, while maintaining a very good accuracy and stability.

We also apply FEM and FDM to the modeling of fuel cells. This is related to one of main problems of civilization, that is the increasing energy hunger combined with limited resources and decreasing reserves. One promising solution to this problem could be the usage of fuel cells, i.e., those devices that convert chemically bounded energy (e.g. hydrogen) directly into electricity. Mathematical description of fuel cells is based on a system of parabolic time-dependent partial differential equations with nonlinear source terms. By adopting appropriate computational techniques to solve this problem, we get an efficient tool to reliably model important phenomena in such devices, thus avoiding expensive measurements and testing instruments.

A different research topic addressed by the Computational Mathematics group consists of developing rigorous computational techniques to study finite and infinite-dimensional systems, like system of ODEs, PDEs, or delay equations. Proving the existence of solutions for nonlinear dynamical system using analytical technique is a very challenging problem, especially in the infinite-dimensional case. New results have been obtained in the last decades following the so-called computer-assisted approach: the combination of analytical tools with rigorous numerics allows to prove the existence of stationary solutions, periodic orbits, connecting orbits, and traveling waves. These dynamical objects are the first ingredients towards the full understanding of the global properties of a nonlinear system, like the existence of chaotic dynamics.

One of the main purposes in computational anatomy is the measurement and statistical study of anatomical variations in organs, notably in the brain or the heart. Several important shape evolution equations that are now used routinely in applications have emerged over time, with the intention of model registration, segmentation or tracking medical images of human organs. In this context our group is studying mesh-free numerical solutions to the Euler-Poincare PDE on diffeomorphisms (EPDiff) as a model to uniquely define the registration of images, and its different extensions to allow not only geometrical changes on shapes representing organs, but the appearance of tumors, plasticity, and the absorption of anatomical structures.
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