Optimized Schwarz Methods without Overlap for the Helmholtz Equation

Abstract

Abstract. The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps. These conditions are, however, nonlocal in nature, and we introduce local approximations which we optimize for performance of the Schwarz method. This leads to an algorithm in the class of optimized Schwarz methods. We present an asymptotic analysis of the optimized Schwarz method for two types of transmission conditions, Robin conditions and transmission conditions with second order tangential derivatives. Numerical results illustrate the effectiveness of the optimized Schwarz method on a model problem and on a problem from industry.Key words. optimized Schwarz methods, domain decomposition, preconditioner, iterative parallel methods, acoustics AMS subject classifications. 65F10, 65N22PII. S10648275013870121. Introduction. The classical Schwarz algorithm has a long history. It was invented by Schwarz more than a century ago [25] to prove existence and uniqueness of solutions to Laplace's equation on irregular domains. Schwarz decomposed the irregular domain into overlapping regular ones and formulated an iteration which used only solutions on regular domains and which converged to a unique solution on the irregular domain. A century later the Schwarz method was proposed as a computational method by Miller in [23], but it was only with the advent of parallel computers that the Schwarz method really gained popularity and was analyzed in depth both at the continuous level (see, for example, [17] [6]. We study in this paper the influence of the transmission conditions on the Schwarz algorithm for the Helmholtz equation. We derive optimal transmission conditions which lead to the best possible convergence of the Schwarz algorithm and which do not require overlap to be effective as in [12]. These optimal

An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

“…Previous optimized Schwarz methods with Robin transmission conditions reduced the number of free parameters by setting s 12 = s 21 . This led in [14,22,21] to an optimized Schwarz method with asymptotic convergence factor…”

“…. .. Table I shows the number of iterations needed for different values of the mesh parameter h for one-sided optimized Robin conditions (see [14,22]), and the new two-sided optimized Robin conditions (see Theorem 4.1), and compares the results with Taylor conditions (i.e. s 12 = s 21 = iω, see [8]) in the case of Krylov acceleration (without, Taylor conditions do not lead to a convergent algorithm, because for all frequencies k > ω, the convergence factor equals 1).…”

“…The name optimized Schwarz methods was introduced in [13] to denote the class of Schwarz methods with improved transmission conditions that has been developed over the previous years in [4,17,20]; for an up to date historical review, and complete results for the positive definite case, see [15]. For Helmholtz problems, optimized Schwarz methods were studied and analyzed with one-sided Robin transmission conditions, and second order transmission conditions in [14,22,21]. A different approach using perfectly matched layers was proposed in [27].…”

Approximation of Optimal Interface Boundary Conditions for Two-Lagrange Multiplier FETI Method

“…The analysis upon the domain size reported Figure 5 show the respective dependence of the methods. The asymptotic behavior of the pro- are still less efficient than those obtained with a continuous approach, see Gander et al [2002], but the implementation of the previous method doesn't depends on a priori knowledge of the problem to be solved (coefficients of the partial differential equation, mesh size, . .…”

“…Interface boundary conditions are the key ingredient to design efficient domain decomposition methods, see Chevalier and Nataf [1998], Benamou and Després [1997], Gander et al [2002]. However, convergence cannot be obtained for any method in a number of iterations less than the number of subdomains minus one in the case of a one-way splitting.…”