Domain decomposition algorithms for parabolic partial differential equations

Domain decomposition methods are considered to solve large systems of equations arising from the discretization of parabolic partial differential equations by finite difference methods. Domain decomposition methods can be classified into two classes, overlapping and non-overlapping methods with respect to the decomposition of the domain. We focus on non-overlapping unconditionally stable domain decomposition methods for solving parabolic partial differential equations with Dirichlet boundary conditions, Dirichlet-Neumann boundary conditions, and pure Neumann boundary conditions. In this dissertation, we propose an Implicit Prediction and Implicit Correction (IPIC) method, a Modified Implicit Prediction (MIP) method, and a Modified Alternating Direction Implicit (MADI) method. In order to maintain unconditional stability, the IPIC method uses a correction procedure on the interface line. However, the correction procedure is not necessary in the MIP method, which is proved in the dissertation. The MADI method is considered as the extreme case of the domain decomposition method without interior points being solved. Numerical results with five different model problems show that the IPIC, MIP, and MADI methods are unconditionally stable and efficient.