Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust, wave number independent convergence and scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.
翻译:在计算上存在各种不同的Helmholtz问题的数字解决办法,其描述性理论在很多流行方法中仍然遥不可及。强健和可伸缩性是大规模应用中实际和可靠的解决问题者的关键,特别是大型波数问题。在这项工作中,我们探索如何使用GenEO型粗皮空间来建立一个适用于高度无限期的Helmholtz问题的两级添加法Schwarz方法。通过对2D型模型问题进行一系列数字测试,并用无污染 meshes的有限元素分解,我们观察到了我们方法的强健、波数独立趋同和可伸缩性。我们进一步提供了结果,表明与DtN粗皮空间进行了有利的比较。我们的数字研究表明,我们的解毒方法可以有效地挑战多种应用。