Available online at www.prace-ri.eu Partnership for Advanced Computing in Europe A Parallel Fast BEM for the Helmholtz Equation as an Extension of SPECFEM3D

In this paper, a new parallel acoustic simulation package has been created, using the boundary element method (BEM). The package is built on top of SPECFEM3D, which is parallel software for doing seismic simulations, e.g. earthquake simulations of the globe. The acoustical simulation relies on a Fourier transform of the seismic elastodynamic data, resulting from SPECFEM3D_GLOBE, which are then postprocessed by a sequence of solutions to Helmholtz equations, in the exterior of the globe. For the acoustic simulations BEM has been employed, which reduces computation to the sphere; however, its naive implementation suffers from quadratic time and memory complexity, with respect to the number of unknowns. To overcome the latter, the method was accelerated by using hierarchical matrices and adaptive cross approximation techniques, which is referred to as Fast BEM. First, a hierarchical clustering of the globe surface triangulation is performed. The arising cluster pairs decompose the fully populated BEM matrices into a hierarchy of blocks, which are classified as far-field or nearfield. While the near-field blocks are kept as full matrices, the far-field blocks are approximated by low-rank matrices. This reduces the quadratic complexity of the serial code to almost linear complexity, i.e. O(n*log(n)), where n denotes the number of triangles. Furthermore, a parallel implementation was done, so that the blocks are assigned to concurrent MPI processes with an optimal load balance. The processes share the triangulation data. The parallel code reduces the computational complexity to O(n*log(n)/N), where N denotes the number of processes. This is a novel implementation of BEM that overcomes computational times of traditional volume discretization methods, e.g. finite elements, by an order of magnitude.