ABSTRACT
This chapter describes direct methods based on the Schur complement and domain decomposition. The coefficient matrix are assumed to be symmetric positive definite. The chapter focuses on iterative methods based on P-regular splittings and domain decompositions. A convergence analysis are examined via the minimization of the equivalent quadratic functional. The chapter presents a message passing interface version of successive over relaxation using domain decomposition and discuses alternating direction implicit (ADI) iterative methods. It aims to define Gauss elimination methods and symmetric positive definite matrices with examples and theorems. ADI iterative methods can be used to approximate the solution to a two variable Poisson problem by a sequence of ordinary differential equations solved in the x or y directions.
