Adaptive finite element analysis of nonlinear problems: balancing of discretization and iteration errors

This article continues prior work of the authors on the combined a posteriori analysis for the discretization and iteration errors in the finite element approximation of linear elliptic problems to the nonlinear case. The underlying theoretical framework is again that of the Dual Weighted Residual (DWR) method for goal-oriented error control. The accuracy in the algebraic solution process can be balanced with that due to discretization using computable a posteriori error estimates in which the outer nonlinear and inner linear iteration errors are separated from the discretization error. This results in effective stopping criteria for the algebraic iteration, which are elaborated particularly for Newton-type methods. The performance of the proposed strategies is demonstrated for several nonlinear test problems.