Abstract of Thesis presented at COPPE/UFRJ as a partial fulfillment of the requirements for the degree of Doctor of Science (D.Sc.)

The Multilevel Schwarz Shooting Method for the Numerical Solution of the Poisson Equation

Christian Emilio Schaerer Serra

July/2002

Advisor:

Eugenius Kaszkurewicz

Department:

Eletrical Engineering

      In this work, the Shooting method, historically used to solve numerically Ordinary Differential Equations, is developed for the numerical solution of the bidimensional Poisson equation subject to Dirichlet boundary conditions. The Simple and Multiple Shooting methods are formulated both by using a feedback control perspective and the Schwarz method was adequately combined in order to introduce overlapping regions. It conduces to an iterative Schwarz - shooting method with low order complexity (per iteration) in terms of floating point operations and memory requirements.
      For accelerating this method, a Multilevel approach is combined adequately. This conduces to an iterative Multilevel Schwarz - Shooting method which presents an opti­mal rate of convergence and complexity, and it is scalable in terms of the problem size. Comparison results between the proposed method and the state of the art method, in this case the Multigrid method, shows that the proposed method is superior in CPU time and number of iterations and it could be considered one-of the fastest method for the Poisson equation. Convergence conditions for the proposed methods are studied from a new splitting algebraic perspective, and analysed exclusively from the charac­teristics of the coefficient matrix of a linear system of equations. A parallel version of the proposed method was developed and formalized for implementation in a distributed homogeneous computational platform.