Efficient solution of two-dimensional wave propagation problems by Cq-Wavelet BEM: Algorithm and applications

In this paper we consider wave propagation problems in two-dimensional unbounded domains, including dissipative effects, reformulated in terms of space-time boundary integral equations. For their solution, we employ a convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. It is known that one of the main advantages of the CQ-BEMs is the use of the FFT algorithm to retrieve the discrete time integral operators with an optimal linear complexity in time, up to a logarithmic term. It is also known that a key ingredient for the success of such methods is the efficient and accurate evaluation of all the integrals that define the matrix entries associated to the full space-time discretization. This topic has been successfully addressed when standard Lagrangian basis functions are considered for the space discretization. However, it results that, for such a choice of the basis, the BEM matrices are in general fully populated, a drawback that prevents the application of CQ-BEMs to large-scale problems. In this paper, as a possible remedy to reduce the global complexity of the method, we consider approximant functions of wavelet type. In particular, we propose a numerical procedure that, by taking advantage of the fast wavelet transform, allows us on the one hand to compute the matrix entries associated to the choice of wavelet basis functions by maintaining the accuracy of those associated to the Lagrangian basis ones and, on the other hand, to generate sparse matrices without the need of storing a priori the fully populated ones. Such an approach allows in principle the use of wavelet basis of any type and order, combined with CQ based on any stable ordinary differential equations solver. Several numerical results, showing the accuracy of the solution and the gain in terms of computer memory saving, are presented and discussed.