In this paper we are concerned with restricted additive Schwarz with local impedance transformation conditions for a family of Helmholtz problems in two dimensions. These problems are discretized by the finite element method with conforming nodal finite elements. We design and analyze a new adaptive coarse space for this kind of restricted additive Schwarz method. This coarse space is spanned by some eigenvalue functions of local generalized eigenvalue problems, which are defined by weighted positive semi-definite bilinear forms on subspaces consisting of local discrete Helmholtz-harmonic functions from impedance boundary data. We proved that a two-level hybrid Schwarz preconditioner with the proposed coarse space possesses uniformly convergence independent of the mesh size, the subdomain size and the wave numbers under suitable assumptions. We also introduce an economic coarse space to avoid solving generalized eigenvalue problems. Numerical experiments confirm the theoretical results.

翻译：本文研究二维Helmholtz方程族在局部阻抗变换条件下的限制加性Schwarz方法。采用协调节点有限元进行离散化，针对此类限制加性Schwarz方法设计并分析了一种新的自适应粗空间。该粗空间由局部广义特征值问题的若干特征函数张成，这些特征函数定义在由阻抗边界数据生成的局部离散Helmholtz调和函数子空间上的加权半正定双线性型中。我们证明，在适当假设下，采用所提出粗空间的两层混合Schwarz预条件子具有独立于网格尺寸、子域尺寸及波数的均匀收敛性。同时引入一种经济型粗空间以规避求解广义特征值问题。数值实验验证了理论结果。