  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 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 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 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 degrees of
freedom in each element.
