Newton's method for convex nonlinear elliptic boundary value problems

Abstract

Summary. Newton's method is applied to solving the boundary value problem for the equation L u =/(x, u) where L is a linear second order uniformly elliptic operator and ](x, u) is a convex monotone increasing function of u for each point x in the domain D. The Newton iterates are shown to converge uniformly, quadratically and monotonically downward to the solution of the problem. The convergence is independent of the choice for the initial Newton iterate. Numerical results are presented for several problems of physical interest.

The state of the art in the numerical solution of time-varying partial differential equations

“…Others have recently generalized them for linear, variable coefficient PDE's in three spatial dimensions.6,11 These techniques have been shown to be practical and efficient for these problems.7,21 Moreover, for nonlinear static PDE's in two spatial dimensions, these ideas have been extended and shown to be very efficient. 23,25 When (if) these techniques mature, the solution of PDE's in two and even three spatial dimensions will be several orders of magnitude cheaper than it currently is, both in terms of computer time and memory.…”