Non-linear Problems

Abstract

Many real-world problems are governed by non-linear mathematical models. The drying of paint, the weather, or the mixing of fluids are just some examples of non-linear phenomenons. In fact, most of the physical, biological, and chemical processes going on around us everyday are described by more or less non-linear laws of nature. Thus, extensions of the finite element method to non-linear equations are of special interest. In this chapter we study the standard methods for tackling non-linear partial differential equations discretized with finite elements , namely, Newton's method and its simplified variant Piccard, or, fixed-point iteration. 9.1 Piccard Iteration Piccard, or fixed-point, iteration is perhaps the most primitive technique for solving non-linear equations. It is applicable to equations of the form x D g.x/ (9.1) where we for simplicity assume that g is a scalar non-linear function of a single variable x. The basic idea is to take a first rough guess x .0/ at the solution, say, N x, and then to compute successively until convergence, for k D 0; 1; 2; : : :, x .kC1/ D g.x .k/ /