Non-overlapping Spectral Additive Schwarz Method

Yu, Yi

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Non-overlapping Spectral Additive Schwarz Method

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In this dissertation, we develop and analyze two domain decomposition methods for elliptic problems with highly heterogeneous coefficients. Our methods are based on the additive average Schwarz (AAS) method that robust with constant coefficients inside each subdomain. The condition number of the preconditioned system is scaleable with $O(H/h)$. The first method is the minimum energy Schwarz (MES) method, which has the best coarse space with constant extension in each subdomain. The condition number of MES is always smaller than the original AAS method. MES method robust when there is only one small eigenvalue in each subdomain. The second methods are the non-overlapping spectral additive Schwarz (NOSAS) methods based on a low-rank discrete energy harmonic extension in each subdomain. To achieve the low-rank discrete energy harmonic extension, we solve a generalized eigenvalue problem in each subdomain. The NOSAS methods have the coarse space with minimum energy if the number of eigenfunctions is given. The condition number of the NOSAS methods does not depend on the coefficients and is similar to that of the AAS method with constant coefficients, with an appropriate threshold of the eigenvalues. Additionally, the NOSAS methods have good parallelization properties. The size of the coarse problem is equal to the total number of eigenvalues chosen in each subdomain. It is only related to the number of high-permeable islands that touch the subdomains’ interface. We also develop and analyze NOSAS in other discretizations. Hybrid Discontinuous Galerkin discretizations of elliptic problems with heterogeneous coefficients are our primary target. Furthermore, we develop three-level domain decomposition preconditioners based on the NOSAS methods. Finally, we show that NOSAS also works for multiscale discretizations.

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