Large-Scale Noise Simulation in Urban Environments Using a Preconditioned Helmholtz Solver

Abstract

This thesis studies the finite element solution of the time-harmonic Helmholtz equation for acoustic wave propagation in urban environments. The discretized Helmholtz equation gives rise to large, complex-valued, indefinite linear systems, for which standard iterative methods may converge very slowly. The aim of this work is to implement and evaluate a preconditioned GMRES solver for such systems, with focus on solver performance, parameter choices, and practical memory limitations. The solver uses a two-level restricted additive Schwarz preconditioner with a spectral coarse space, following the RAS/MS-GFEM approach of Ma, Alber, Scheichl, and Zhang. The implementation is first verified on an exact plane-wave problem, then tested on a controlled rectangular box domain, and finally applied to realistic city-domain meshes representing part of Gothenburg. The experiments investigate how mesh resolution, frequency, and the choice of coarse space affect convergence and computational cost. The results show that the two-level coarse correction considerably improves GMRES convergence compared with a one-level method and with plain GMRES without preconditioning. On the box domain, the method gives stable algebraic convergence up to 30 Hz, although the acoustic field is not fully mesh-converged at the higher frequencies. On the city geometry, the 5 Hz case gives the clearest mesh-refinement behaviour, while the 10 Hz, 20 Hz, and 30 Hz cases become increasingly demanding. The largest city runs show that explicit storage of the coarse basis can become memory-limiting. The study therefore shows that the two-level spectral Schwarz preconditioner is promising for large Helmholtz problems in urban acoustics, but that higher-frequency simulations require careful choices of mesh resolution, coarse-space size, and memory-efficient implementation. Keywords: