Solving large nonlinear systems of equations by an adaptive condensation process (Q1104703)

Let A be a symmetric positive definite \(m\times m\)-matrix and F a mapping acting from \({\mathbb{R}}^ m \)into \({\mathbb{R}}^ m \)with symmetric Fréchet derivative. The authors present an algorithm which efficiently solves large nonlinear systems of equations of the form (*) \(Ax=F(x)\). Such systems arise from the discretization of nonlinear elliptic partial differential equations. By means of an adaptive decomposition process the system (*) is split into a low dimensional and a remaining high dimensional system. To solve the first one a sophisticated Newton-type method is presented. The other system is said to be easily solved by fixed point iteration. A convergence analysis is given. As a numerical example the common 5-point finite difference discretization of a nonlinear Dirichlet problem in two dimensions containing a real parameter is considered.