Maksymilian Dryja | Searcharxiv

arXiv2021

From Additive Average Schwarz Methods to Non-overlapping Spectral Additive Schwarz Methods

In this paper, we design and analyze two new methods based on additive average Schwarz -- AAS method introduced in \cite{MR1943457}. The new methods design for elliptic problems with highly heterogeneous coefficients. The methods are of the non-overlapping type, and the subdomain interactions obtain 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 non-overlapping 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 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. It is only related to the number of high-permeable islands that touch the subdomains' interface.

math.NA↗

arXiv2014

The analysis of FETI-DP preconditioner for full DG discretization of elliptic problems

In this paper a discretization based on discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region $Ω$ which is a union of $N$ disjoint polygonal subdomains $Ω_i$ of diameter $O(H_i)$. The discontinuities of the coefficients, possibly very large, are assumed to occur only across the subdomain interfaces $\partial Ω_i$. In each $Ω_i$ a conforming quasiuniform triangulation with parameters $h_i$ is constructed. We assume that the resulting triangulation in $Ω$ is also conforming, i.e., the meshes are assumed to match across the subdomain interfaces. On the fine triangulation the problem is discretized by a DG method. For solving the resulting discrete system, a FETI-DP type method is proposed and analyzed. It is established that the condition number of the preconditioned linear system is estimated by $C(1 + \max_i \log H_i/h_i)^2$ with a constant $C$ independent of $h_i$, $H_i$ and the jumps of coefficients. The method is well suited for parallel computations and it can be extended to three-dimensional problems. This result is an extension, to the case of full fine-grid DG discretization, of the previous result [SIAM J. Numer. Anal., 51 (2013), pp.~400--422] where it was considered a conforming finite element method inside the subdomains and a discontinuous Galerkin method only across the subdomain interfaces. Numerical results are presented to validate the theory.

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