Analysis of a Multilevel Iterative Method for Nonlinear Finite Element Equations

The multilevel iterative technique is a powerful technique for solving systems of equations associated with discretized partial diierential equations. We describe how this techniques can be combined with a globally con-vergent approximate Newton method to solve nonlinear partial diierential equations. We show that asymptotically only one Newton iteration per level is required; thus the complexity for linear and nonlinear problems is essentially equal. 1. Introduction In this discussion we present an extension of a multilevel iterative method for linear elliptic equations to nonlinear elliptic boundary value problems. In particular, we show how to use an approximate-Newton multilevel scheme to solve discrete nonlinear systems of equations which arise from a standard weak formulation of the nonlinear partial diierential equation. The framework of our analysis combines the multilevel iterative methods for linear nite element equations discussed in Bank and Dupont 3] and Bank 2] with the global approximate Newton setting of Bank and Rose 5, 4]. Under appropriate conditions of elliptic regularity, we show that both the continuous and discrete solutions exist and that our scheme converges to an approximation within the dis-cretization error of the continuous problem in time (and also space) proportional