Solution of nonlinear Poisson-type equations

Abstract

We consider approximate and truncated Newton methods for solving nonlinear Poisson-type systems of equations with emphasis on vector computers. We have selected two model differential equations for our analysis. The first, after finite difference discretization, yields a nonlinear system with a symmetric positive definite Jacobian matrix, while the second yields a nonsymmetric Jacobian matrix which is not, in general, positive definite. In the case of a symmetric positive definite Jacobian matrix, the linearized Newton systems are solved by (preconditioned) conjugate gradient methods. In particular, we consider incomplete Cholesky factorization in conjunction with the red-black ordering scheme and symmetric diagonal scaling as preconditioners. In the case of a nonsymmetric Jacobian matrix, we construct symmetric positive definite approximations and analyze the effect of these approximations. We develop, and present theory for, a truncated approximate Newton method. This method uses an approximate Jacobian matrix in addition to truncation in the solution of the Newton system. Convergence properties and work reduction of this new method have been found to be very competitive.