Two-level overlapping Schwarz methods based on local generalized eigenproblems for Hermitian variational problems

The research of two-level overlapping Schwarz (TL-OS) method based on constrained energy minimizing coarse space is still in its infancy, and there exist some defects, e.g. mainly for Poisson-like problems and too heavy computational cost of coarse space construction. In this paper, by introducing appropriate assumptions, we propose more concise coarse basis functions for general Hermitian positive and definite discrete systems, and establish the corresponding TL-OS preconditioned algorithmic and theoretical frameworks. Furthermore, to enhance the practicability of the algorithm, we design two economical TL-OS preconditioners and prove that the condition number is robust with respect to the model and mesh parameters. As the first application of the frameworks, we prove that the assumptions hold for the linear finite element discretization of Poisson problem with high contrast and oscillatory coefficient. In particular, we also prove that the condition number of the economically preconditioned system is independent of the jump range under a certain jump distribution. Experimental results show that the first kind of economical preconditioner is more efficient and stable than the existed one. Secondly, we construct TL-OS and the economical TL-OS preconditioners for the plane wave least squares discrete system of Helmholtz equation by using the frameworks. The numerical results for homogeneous and non-homogeneous cases illustrate that the PCG method based on the proposed preconditioners have good stability in terms of the angular frequency, mesh parameters and the number of freedoms in each element.