A Cousin Formulation for Overlapped Domain Decomposition Applied to the Poisson Equation

Abstract

In this paper we present a domain decomposition method for the solution of a linear equation as a direct two-step procedure (in contrast to iterativebased Schwarz and Aitken-Schwarz procedures): first, local solutions to the nonhomogeneous equation are calculated on overlapping domains, and second, connection functions, solutions to the homogeneous equation, are calculated which satisfy the Cousin problem and correct for the initial solution mismatch at interior boundaries. The procedure is applied to Poisson's equation as a sample linear equation. The calculation of the connection functions is achieved through a classical orthogonal decomposition of the solution to Laplace's equation and can be achieved through progressive solutions in each direction for separable boundary conditions. The procedure is also applicable to more complicated domains with non-separable boundary conditions. A few examples will be given. The connection functions can be calculated to high precision and require minimal overlap of the subdomains.