Adaptive CEM-TL-OS Preconditioner For Hermite Variational Problems And Its Applications

Domain decomposition method(DDM)is one of the important methods to construct preconditioners for discrete systems of partial differential equations.In recent years,the adaptive domain decomposition method based on constrained energy minimizing coarse spaces(CEM-DDM)is a research hotspot on DDM.It is widely concerned because the controllable condition number of the adaptive CEM-DDM.Compared with the non-overlapping method,the research on the overlapping CEM-DDM method is still in its infancy.This dissertation focus on the adaptive two-level overlapping Schwarz method based on constrained energy minimizing coarse space(adaptive CEM-TL-OS).For general Hermitian and positive definite discrete systems,two kinds of coarse basis functions of constraint energy minimizing space are constructed by introducing some assumptions and local generalized eigenvalue problems.We establish the algorithmic and theoretical frameworks of the corresponding CEM-TL-OS preconditioners,including a new class of basis coarse functions which has a more concise form and lower computational complexity.Since the computation of the above coarse basis functions involve the entire domain,two more practical and economical adaptive CEM-TL-OS preconditioners are designed by restricting the solution domain to its subdomains,and the theories of the condition number estimation are established under stronger assumptions.Especially we develop a new estimation theory for the second preconditioner.For the linear finite element discretization of a second order elliptic problem,we construct two adaptive CEM-TL-OS preconditioners and their economical forms by using the above algorithmic and theoretical frameworks,and we also establish the estimation theories of the condition number.In particular,we prove that the condition number of the economically preconditioned system is independent of the jump range when the coefficient p(x)satisfies a certain jump distribution.Furthermore,by using the generalized eigenvalue problem,the finite element knowledge,and spectral equivalence technique,we propose a stability estimation of the key factor Cp(OM)for the estimation of the condition number,which is more consistent with the numerical results.In addition,for the second-order elliptic problem without zero-order term,we propose another estimation of the condition number based on the "estimation theory of reduced condition number".Moreover,we design a fast algorithm by essentially transforming the generalized eigenvalue problem into two ordinary eigenvalue problems.Numerical experiments verify the correctness of the theoretical results.It shows that the new economical preconditioner is more robust with respect to the grid size,the overlapping width,the jump coefficient distribution and the jump range,and the computational efficiency is greatly improved.For the plane wave least squares discrete systems of the homogeneous and non-homogeneous Helmholtz equations,under the designed algorithmic and theoretical frameworks,we construct the adaptive CEM-TL-OS preconditioner and its economical form by introducing appropriate bilinear functional and partition of unity functions defined on the dual mesh.In particular,by designing a special solution problem,the non-homogeneous problems is homogenized and the computational efficiency is improved.The numerical results show that the preconditioned conjugate gradient(PCG)method based on the proposed preconditioners have good robustness of the angular frequency,the mesh size,the number of degrees of freedom in each element and the overlapping width.