Maria Lopez-Fernandez

Jose M. Arrieta, Maria Lopez-Fernandez, Enrique Zuazua

We consider an evolution equation of parabolic type in R having a travelling wave solution. We perform an appropriate change of variables which transforms the equation into a non local evolution one having a travelling wave solution with zero speed of propagation with exactly the same profile as the original one. We analyze the relation of the new equation with the original one in the entire real line. We also analyze the behavior of the non local problem in a bounded interval with appropriate boundary conditions and show that it has a unique stationary solution which is asymptotically stable for large enough intervals and that converges to the travelling wave as the interval approaches the entire real line. This procedure allows to compute simultaneously the travelling wave profile and its propagation speed avoiding moving meshes, as we illustrate with several numerical examples.

Steffen Börm, Maria Lopez-Fernandez, Stefan Sauter

The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated; hence their sparse approximation is of outstanding importance. In our paper we will generalize the directional $\mathcal{H}^{2}$-matrix techniques from the \textquotedblleft pure\textquotedblright\ Helmholtz operator $\mathcal{L}u=-Δu+ζ^{2}u$ with $ζ=-\operatorname*{i}k$, $k\in\mathbb{R}$, to general complex frequencies $ζ\in\mathbb{C}$ with $\operatorname{Re}ζ>0$. In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition which contains $\operatorname{Re}ζ$ in an explicit way and introduce the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent \textit{directional expansion functions}. We develop an error analysis which is explicit with respect to the expansion order and with respect to $\operatorname{Re}ζ$ and $\operatorname{Im}ζ$. This allows to choose the \textit{variable }expansion order in a quasi-optimal way depending on $\operatorname{Re}ζ$ but independent of, possibly large, $\operatorname{Im}ζ$. The complexity analysis is explicit with respect to $\operatorname{Re}ζ$ and $\operatorname{Im}ζ$ and shows how higher values of $\operatorname{Re}% ζ$ reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its nearfield part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation.

Maria Lopez-Fernandez

The time integration of semilinear parabolic problems by exponential methods of different kinds is considered. A new algorithm for the implementation of these methods is proposed. The algorithm evaluates the operators required by the exponential methods by means of a quadrature formula that converges like $O(e^{-cK/\ln K})$, with $K$ the number of quadrature nodes. The algorithm allows also the evaluation of the associated scalar mappings and in this case the quadrature converges like $O(e^{-cK})$. The technique is based on the numerical inversion of sectorial Laplace transforms. Several numerical illustrations are provided to test the algorithm.