Best Robin Parameters for Optimized Schwarz Methods at Cross Points

Abstract

Optimized Schwarz methods are domain decomposition methods in which a largescale PDE problem is solved by subdividing it into smaller subdomain problems, solving the subproblems in parallel, and iterating until one obtains a global solution that is consistent across subdomain boundaries. Fast convergence can be obtained if Robin conditions are used along subdomain boundaries, provided that the Robin parameters p are chosen correctly. In the case of second order elliptic problems such as the Poisson equation, it is well known for two-subdomain problems without overlap that the optimal choice is p = O(h −1/2) (where h is the mesh size), with the resulting method having a convergence factor of ρ = 1 − O(h 1/2). However, when cross points are present, i.e., when several subdomains meet at a single point, this choice leads to a divergent method. In this article, we show for a model problem that convergence can only occur if p = O(h −1) at the cross point; thus, a different scaling of the Robin parameter is needed to ensure convergence. In addition, this choice of p allows us to recover the 1 − O(h 1/2) convergence factor in the resulting method.

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

“…At the discrete level, a condition number estimate for a finite element discretizations of isotropic diffusion problems and Robin transmission conditions can be found in [33]. Two different consistent discretizations at cross points for finite element discretizations were derived and analyzed in [22], and optimized Robin parameters at cross points at the algebraic level were derived in [20], but classical energy estimates can not directly be used in the presence of cross points [21]. There is also to the best of the authors knowledge no study so far on efficient coarse spaces for anisotropic diffusion problems.…”

On the Applicability of Lions’ Energy Estimates in the Analysis of Discrete Optimized Schwarz Methods with Cross Points

“…Note that neither formulation is an exact discretization of (2) at cross points; thus, Lions' convergence analysis does not apply there. In fact, one can show [4] that the eigenvalues of the iteration matrix of the 2LM method may lie outside the unit disc when cross points are present, as seen in the 4 × 4 example shown in Figure 3. In such cases, the method would diverge.…”

“…In such cases, the method would diverge. However, convergence can be restored if one uses Robin parameters with a different scaling at the cross points [4].…”

Multigrid technique and Optimized Schwarz method on block-structured grids with discontinuous interfaces

“…In the current method the weighting coefficient in Exp. (25) is chosen to be equal to 1/2. To ensure a high value of the Robin parameter at block corners on the discontinuous interface the parameter B k with highest value may be weighted with a higher weighed coefficient, thus preserving the O(h −1 ) scaling of the Robin parameter at block corners not only at internal continuous block boundaries, but also at any block corner position of internal discontinuous interfaces.…”