A domain decomposition method with fast convergence for the Helmholtz equation

Solving the Helmholtz equation by finite element methods is quite important in acoustics. When the frequency or the size of the problem increase, large meshes are necessary and consequently heavy computations are required. One possibility is to use domain decompositions for which the domain is decomposed into subdomains on which the solutions can be computed more easily. This involves an iterative scheme where data are transmitted between subdomains from the precedent iteration. The main problem is to have a low number of iterations so that the problem can be solved in a reasonable amount of time. In this work, we present a domain decomposition method based on two main features. The first one is to use extended domains with absorbing boundary conditions. The second feature is to decompose the whole domain into one-dimensional or two-dimensional networks of subdomains so that double sweep preconditioners can be used. Examples are shown where the number of iterations is usually low. This number of iterations is also shown to depend slowly on the number of domains and the frequency.