The construction of efficient solvers for non self-adjoint problems, like Helmholtz equations is a challenging task. After the discretisation of the PDE by a finite element method, the resulting linear systems are large and because of their spectral properties, difficult to analyse theoretically and to solve by iterative methods. Domain decomposition methods are hybrid methods, as they use an iterative coupling of smaller problems which are solved in turn by direct methods. They rely on dividing the global problem into local subproblems on smaller subdomains.
These methods can be used as iterative solvers but also as preconditioners in a Krylov method. Robustness with respect of the number of subdomains is important as this is related to the notion of scalability. We focus here on a configuration where scalability is achieved without the addition of a coarse-space correction. However, convergence can still be improved by modifying the transmission conditions imposed between the subdomains.
In this manuscript, we start by giving an overview of the basic domain decomposition methods and their use as preconditioners. Then we consider these methods from an iterative point of view and we perform a study of convergence analysis of overlapping Schwarz methods with Dirichlet, Robin, zeroth and second order transmission conditions for many subdomains. We also present more sophisticated methods, which implement more effective transmission conditions depending on some optimised parameters. In our analysis, we focus on the Helmholtz problem and the magnetotelluric approximation of Maxwell’s equation for stripwise decompositions into many domains. Our theoretical findings are being demonstrated by the appropriate numerical evidence.