Iterative solution with shifted Laplace preconditioner for plane wave enriched isogeometric analysis and finite element discretization for high-frequency acoustics

In this paper we investigate the iterative solution of enriched finite element methods for solving a frequency domain wave problem. The considered methods are partition of unity isogeometric analysis (PUIGA) and the partition of unity finite element method (PUFEM). We study the performance of an operator based preconditioner, namely, the shifted Laplace preconditioner against a complex shifted ILU preconditioner. Compared to complex shifted ILU, the shifted Laplace preconditioner leads to a significant performance improvement in terms of reduced number of GMRES iterations. Through numerical examples, we show that the shifted Laplace preconditioner results in a wavenumber independent GMRES convergence for the frequency range considered. We also show in general that preconditioned GMRES performs better with PUIGA than for PUFEM. The improvement is explained in terms of the lower condition numbers with PUIGA at higher frequencies. This work is one of the first attempt to evaluate the performance of operator based preconditioners on a Trefftz type method for solving frequency domain wave problems.