Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

Abstract

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré-Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient.

Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

“…The results there are valid for many subdomains. Dubois and Lui [5] discuss an optimized Schwarz method in the case where there are large jumps in the coefficients of the PDE. One example of an optimized Schwarz method using a second order boundary condition along the artificial interface is…”

An Optimal Robin-Type Domain Decomposition Method for Raviart-Thomas Vector Field in Three Dimensions

“…Two-level overlapping Schwarz preconditioners for these spaces have been developed for two (see [3]) and three (see [4][5][6][7][8][9][10]) dimensions, respectively. Multigrid and multilevel methods are considered in [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. A few papers on the ( ; ) H curl Ω case have proved optimality for a two-subdomain iterative sub structuring preconditioner, combined with Richardson's method, for a lowfrequency approximation of time-harmonic Maxwell's equations in three dimensions.…”

Best Robin Parameters for Optimized Schwarz Methods at Cross Points

“…; M is a matrix with ones on the diagonal and M ij = − 1 di−1 whenever i is a point adjacent to d i subdomains and j = i corresponds to the same physical point as i, and G has the same non-zero pattern as M , except all entries are +1. We claim that the discrete formulation (3) is equivalent to the stationary iteration (8) λ…”

“…, where k min and k max are the minimum and maximum frequencies that can be resolved by the spatial grid. For the sake of easy implementation, we have used (33) as a guideline for choosing our parameter p * away from cross points, even though better choices are available for problems with jumps in the coefficients [8]. We calculate the optimal parameter p * for different levels of refinement from the coarse mesh using (33); since k min = C and k max = C ′ /h for some constants C and C ′ , we have p * = O(1/ √ h) for η fixed and h small enough.…”