Two-level Overlapping Schwarz Methods Based on Local Generalized Eigenproblems for Hermitian Variational Problems

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The research of the two-level overlapping Schwarz (TL-OS) methods based on constrained energy minimizing coarse space is still in its infancy, and there exist some defects, e.g., mainly for a second order elliptic problem 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 algorithmic and theoretical frameworks of the corresponding TL-OS methods. Furthermore, to enhance the practicability of the algorithm, we design two economical TL-OS preconditioners and prove the condition number estimate. As the first application of the frameworks, we prove that the assumptions hold for the linear finite element discretization of a second order elliptic problem with high contrast and oscillatory coefficient and the condition number of the TL-OS preconditioned system is robust with respect to the model and mesh parameters. 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 existing one. Second, we construct TL-OS and the economical TL-OS preconditioners for the plane wave least squares discrete system of the Helmholtz equation by using the frameworks. The numerical results for homogeneous and nonhomogeneous cases illustrate that the preconditioned conjugate gradient method based on the proposed preconditioners has good stability in terms of the angular frequency, mesh parameters, and the number of degrees of freedom in each element.