Solving large nonlinear systems of equations by an adaptive condensation process

SummaryWe present an algorithm which efficiently solves large nonlinear systems of the form $$Au = F(u), u \in \mathbb{R}^n $$ whenever an (iterative) solver “A−1” for the symmetric positive definite matrixA is available andF'(u) is symmetric. Such problems arise from the discretization of nonlinear elliptic partial differential equations. By means of an adaptive decomposition process we split the original system into a low dimensional system — to be treated by any sophisticated solver — and a remaining high-dimensional system, which can easily be solved by fixed point iteration. Specifically we choose a Newton-type trust region algorithm for the treatment of the small system. We show global convergence under natural assumptions on the nonlinearity. The convergence results typical for trust-region algorithms carry over to the full iteration process. The only large systems to be solved are linear ones with the fixed matrixA. Thus existing software for positive definite sparse linear systems can be used.