A Multiplicative Schwarz Method and Its Application to Nonlinear Acoustic - Structure Interaction

A new Schwarz method for nonlinear systems is presented, constituting the multiplicative variant of a straightforward additive scheme. Local convergence can be guaranteed under suitable assumptions. The scheme is applied to nonlinear acoustic-structure interaction problems. Numerical examples validate the theoretical results. Further improvements are discussed by means of introducing overlapping subdomains and employing an inexact strategy for the local solvers. Mathematics Subject Classification. 74F10, 65B99, 65M12. Received November 20, 2007. Revised August 14, 2008. Published online April 8, 2009. Introduction We present a new Schwarz type domain decomposition method for nonlinear problems. Such methods typically lead to schemes where an outer iteration for the subproblem correction and an inner subspace iteration on each subproblem have to be applied [15,17,18]. Our method constitutes, by means of a Gauss-Seidel type outer iteration, a multiplicative variant of the straightforward additive scheme introduced in [7], and further improved in [1,5]. In Section 1, we first recall the general setting and the theoretical results of the additive Schwarz method for nonlinear problems. Based on this framework, we introduce and analyze the corresponding multiplicative version. In the subsequent sections, the domain decomposition scheme is applied to nonlinear acoustic-structure interaction problems. In particular, we extend the elasto-acoustic problem formulation of [8] to geometrically nonlinear structures [6], and nonlinear acoustic wave propagation [13]. After introducing the single field problems in Section 2, we present the coupled problem formulation in Section 3, yielding two numerical schemes using different solvers for the subproblems, a Newton-like and a fixed point iteration scheme. We finally present the results of several numerical examples in Section 4. First, the theoretical results are validated and the convergence behavior of the additive and the multiplicative variant are compared. Moreover, further possible improvements of the scheme are discussed, one by the introduction of a small subregion where both subdomains overlap, and another by employing an inexact strategy for the local linear solver.