Domain decomposition methods for eigenvalue problems

This paper proposes several domain decomposition methods to compute the smallest eigenvalue of linear self-adjoint partial differential operators. Let us be given a partial differential operator on a domain which consists of two nonoverlapping subdomains. Suppose a fast eigenvalue solver is available for each of these subdomains but not for the union of them. The first proposed method is a scheme which determines an appropriate boundary condition at the interface separating the two regions. This boundary condition can be derived from the zero of a nonlinear operator which plays the same role for the eigenvalue problem as the Steklov–Poincaré operator for the linear equation. An iterative method can be used to solve this nonlinear equation, yielding the exact boundary condition at the interface. This enables the determination of the eigenpair on the whole domain. The same concept can be applied to a Schwarz alternating method for the eigenvalue problem in case the subdomains overlap. A nonoverlapping scheme (in the spirit of Schur complement) will also be discussed. These ideas are also applicable to a domain imbedding algorithm.