Application of the Mesh Independence Principle to Mesh Refinement Strategies

The Mesh Independence Principle (MIP) for a Newton iteration applied to an operator equation and its discretization for different “stepsizes” h states for a fixed starting point the number of iterations of Newton’s method to obtain a fixed tolerance is eventually independent of h. This MIP is used to formulate an efficient strategy for the solution of nonlinear equations amounting to the work equivalent of two or three Newton iterates for the smallest h. Examples are given.