From Additive Average Schwarz Methods to Nonoverlapping Spectral Additive Schwarz Methods
Yi Yu, Maksymilian Dryja, Marcus V. Sarkis · SIAM Journal on Numerical Analysis · 2021
In this paper, we design and analyze two new methods based on the additive average Schwarz (AAS) method introduced in [3]. The new methods are designed for elliptic problems with highly heterogeneous coefficients. The methods are of the nonoverlapping type, and the subdomain interactions are obtained via the coarse space. The first method is the minimum energy Schwarz (MES) method. MES has the minimum energy for the coarse space with constant extension inside each subdomain. The condition number of the MES method is always smaller than in the AAS method. The second class of methods is the nonoverlapping spectral additive Schwarz (NOSAS) methods based on low-rank discrete energy harmonic extension in each subdomain. To achieve the low-rank, we solve a generalized eigenvalue problem in each subdomain. NOSAS methods have the minimum energy for a given rank of the coarse space. The condition number of the NOSAS methods does not depend on the coefficients. Additionally, the NOSAS methods have good parallelization properties. The size of the global problem is equal to the total number of eigenvalues chosen in each subdomain. For a class of coefficients, the number of eigenvalues is only related to the number of high-permeable islands that touch the subdomains' interface.