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.

在本文中，我们研究了用于解决频域波动问题的丰富有限元方法的迭代解决方案。所考虑的方法是统一的划分等几何分析 (PUIGA) 和划分单位有限元法 (PUFEM)。我们研究了基于运算符的预处理器的性能，即相对于复杂的移位 ILU 预处理器的移位拉普拉斯预处理器。与复杂的移位 ILU 相比，移位拉普拉斯预处理器在减少 GMRES 迭代次数方面带来了显着的性能改进。通过数值例子，我们表明移位的拉普拉斯预处理器在考虑的频率范围内导致波数独立的 GMRES 收敛。总的来说，我们还表明，经过预处理的 GMRES 在 PUIGA 上的表现比在 PUFEM 上表现得更好。这种改进是通过 PUIGA 在较高频率下的较低条件数来解释的。