Solution of the Exterior Boundary Value Problem for the Helmholtz Equation Using Overlapping Domain Decomposition

A method for solving the exterior three-dimensional boundary value problem for the Helmholtz equation is proposed and investigated. The method is based on an overlapping decomposition of the external domain and on the Schwarz alternating method with the successive solution of the interior and exterior boundary value problem in overlapping subdomains on the adjacent boundaries of which iterated interface conditions are set. Sufficient conditions for the convergence of the method in the case of a negative coefficient in the Helmholtz equation are found. Convergence of a special case of the problem is analyzed, and conclusion on the applicability of the proposed approach to solving problems with an arbitrary wave number is drawn. The proposed method is successfully applied in combination with the finite volume method to the numerical solution of interior boundary value problems and in combination with Green’s formula for solving exterior boundary value problems. The convergence rate of the iterations and the accuracy of computations is illustrated by a series of computational experiments. The choice of decomposition parameters that ensure the convergence of the method is analyzed.