The following document contains the notes prepared for a course to be delivered by the second author at the XVI Jacques-Louis Lions Spanish-French School on Numerical Simulation in Physics and Engineering, in Pamplona (Spain), September 2014. We will not spend much time with the introduction. Let it be said that this is a course on how to approximate causal convolutions and convolution equations, that is, expressionswhere either g or h is unknown. This seems like a very small problem to be working on when it is presented in this flippant form. The truth is hidden in what the convolution integral means. We will be dealing with operator valued distributions f in convolution with a function valued distribution g. A large set of problems related to the scattering of linear waves (acoustic, elastic, electromagnetic) can be written in this form after being moved to the boundary of the scatterer. We will focus on acoustic waves (we will actually restrict all our attention to a single model problem) and on a particular class of discretization methods, the so-called Convolution Quadrature method, introduced by Christian Lubich in the mid-eighties.These notes will emphasize the introduction of concepts and algorithms in a rigorous language, while avoiding proofs. As a matter of fact, we will not state a single theorem explicitly. (This is not due to us not liking theorems, but with the goal of keeping a more narrative tone.) However, clear results will be stated as part of the text.The mathematics of the field of time domain boundary integral equations (which include many important and highly non-trivial examples of convolution equations) involve many interesting and deep analytic concepts, as the reader will be able to ascertain from these notes. The deeper mathematical structure of this field is explored in the lecture notes [28], in a step-by-step traditional mathematical fashion, with no much time for computation. From that point of view, these notes represent the algorithmic counterpart to [28]. Convolution Quadrature is not the only method to approximate convolution equations that appear in wave propagation phenomena. Galerkin and collocation methods compete with CQ in interest, applicability, and good properties. We will not discuss or compare methods, especially because much is still to be explored both in theory and 1 arXiv:1407.0345v1 [math.NA] 1 Jul 2014 practice. We will not comment on existing literature on CQ for scattering problems on elastic and electromagnetic waves either.Before we start, let us take some time for acknowledgements. Our research is partially funded by the National Science Foundation (grant DMS 1216356). We now become I. I (FJS) want to thank the organizers of the EHF2014 for the invitation to participate in the school. It is actually my second time in this series (the first one was in Laredo, so many years ago). Since then, the school has made its name even longer by honoring the extraordinary Jacques-Louis Lions, who happens to be my academic great-grandfather. Much of what I know on tim...

We present a novel analysis of the boundary integral operators associated to the wave equation. The analysis is done entirely in the time-domain by employing tools from abstract evolution equations in Hilbert spaces and semi-group theory. We prove a single general theorem from which well-posedness and regularity of the solutions for several boundary integral formulations can be deduced as particular cases. By careful choices of continuous and discrete spaces, we are able to provide a concise analysis for various direct and indirect formulations, both at the continuous level and for their Galerkin-in-space semi-discretizations. Some of the results here are improvements on previously known results, while other results are equivalent to those in the literature. The methodology presented here greatly simplifies the analysis of the operators of the Calderón projector for the wave equation and can be generalized for other relevant boundary integral equations.

We study a symmetric BEM-FEM coupling scheme for the scattering of transient acoustic waves by bounded inhomogeneous anisotropic obstacles in a homogeneous field. An incident wave in free space interacts with the obstacles and produces a combination of transmission and scattering. The transmitted part of the wave is discretized in space by finite elements while the scattered wave is reduced to two fields defined on the boundary of the obstacles and is discretized in space with boundary elements. We choose a coupling formulation that leads to a symmetric system of integro-differential equations. The retarded boundary integral equations are discretized in time by Convolution Quadrature, and the interior field is discretized in time with the trapezoidal rule. We show that the scattering problem generates a C 0 group of isometries in a Hilbert space, and use associated estimates to derive stability and convergence results. We provide numerical experiments and simulations to validate our results and demonstrate the flexibility of the method.

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