Optimized Schwarz Methods for Circular Domain Decompositions with Overlap

Abstract

Abstract. Optimized Schwarz methods are based on transmission conditions between subdomains which are optimized for the problem class that is being solved. Such optimizations have been performed for many different types of partial differential equations, but were almost exclusively based on the assumption of straight interfaces. We study in this paper the influence of curvature on the optimization, and we obtain four interesting new results: first, we show that the curvature does indeed enter the optimized parameters and the contraction factor estimates. Second, we develop an asymptotically accurate approximation technique, based on Turán type inequalities in our case, to solve the much harder optimization problem on the curved interface, and this approximation technique will also be applicable to currently too complex best approximation problems in the area of optimized Schwarz methods. Third, we show that one can obtain transmission conditions from a simple circular model decomposition which have also been found using microlocal analysis but that these are not the best choices for the performance of the optimized Schwarz method. Finally, we find that in the case of curved interfaces, optimized Schwarz methods are not necessarily convergent for all admissible parameters. Our optimization leads, however, to parameter choices that give the same good performance for a circular decomposition as for a straight interface decomposition. We illustrate our analysis with numerical experiments.Key words. optimized Schwarz method, optimized transmission conditions, circular domain decomposition, interface curvature, parallel computation AMS subject classifications. 65N55, 65F10DOI. 10.1137/130946125 1. Introduction. Domain decomposition methods are important techniques for the numerical simulation of large scale physical problems, since they can systematically reduce their complexity by decomposition and lead to efficient parallel solvers. Among the many domain decomposition methods, optimized Schwarz methods, going back to an idea for a nonoverlapping method by Lions [28], have attracted substantial attention over the past decade because their transmission conditions can be adapted to the physics of the underlying problem and thus lead to very efficient methods for hard problems; for an overview and references, see [35,17]. Optimized Schwarz methods are a very active area of research, and they have found their way into many areas of applications: for Helmholtz problems, see [12,6,22], for advection diffusion evolution problems, see [34,20,7,40], for wave equations, see [21,19], for Maxwell problems, see [41,13,38,37,8], and for shallow water and the primitive equations, see [39,3]. Optimized Schwarz methods have also led to new theoretical developments (see, for example, [27,30,29]) and many innovative preconditioners that appeared under different names; see, for example, the sweeping preconditioner in [16,2,23,14], the source transfer method in [11], and also [42], which are among the best iterative

The influence of domain truncation on the performance of optimized Schwarz methods

“…However, unlike in the infinite domain decomposition analysis [14], the above min-max problem can not be solved directly. Here, we use the technique applied in [20,23], that is to say, when the frequency k is small, we use the exact convergence factor ρ OO0 directly; when the frequency k is large, we use the approximate convergence factor ρ app instead of the exact convergence factor. We can prove that we asymptotically solve the min-max problem (3.5): THEOREM 3.4 (OO0, overlapping case).…”

“…where OO2 stands for "Optimized of Order 2", a second-order transmission condition that is also known as the optimized Ventcell transmission condition in the literature. Again, we need the technique used in [20,23] for the analysis to obtain the following results. THEOREM 3.6 (OO2, overlapping case).…”

“…Their performance can be optimized by appropriately choosing parameters in the transmission conditions between subdomains, and the methods have received considerable attention over the past decade in the DD literature; for overviews, see [14,35] and the monograph [10]. Optimized Schwarz methods have not only led to many theoretical developments, for example [17,20,22,23,29,32,33], but also have been proven very useful in many applications: for instance, we refer to [3,7,19,24] for Helmholtz problems, [4,16,34,41] for advection diffusion problems, [1,9,11,36,37] for Maxwell's equations, [39,40] for shallow water problems, [2] for primitive equations, [27] for the fluid-structure interaction problem, and [28] for an electrocardiology simulation. Optimized Schwarz methods can be used as efficient preconditioners as well; for example, see [18,30] for an optimized Schwarz preconditioner.…”

“…Without loss of generality, we assume a + L < b; see Figure 1.1. This setting contains many existing results as special cases, e.g., a = b = ∞ leads to the case of an infinite domain decomposition analyzed in [14]; a = b finite leads to the finite symmetric domain decomposition in [15], and if b = ∞ and a is finite, then we have a one-sided bounded domain decomposition, which is similar to a circular domain decomposition; see [20,23].…”

The sub-Cauchy–Stokes problem: Solvability issues and Lagrange multiplier methods with artificial boundary conditions

“…Let us define by e θ , and e r the polar unit vectors. Then, similar to the derivation of the optimized Schwarz methods, we assume that the flow is circular, that is, u = u(r)e θ , so that the streamlines are circular [43,44,45,26]. The incompressibility condition ∇ · u is satisfied by any flow of this form, and the Stokes equations are therefore reduced to ∂p ∂r = 0, 1 < r < a,…”