Holst M., Szypowski R., Zhu Y. (2013) Adaptive Finite Element Methods with Inexact Solvers for the Nonlinear Poisson-Boltzmann Equation. In: Bank R., Holst M., Widlund O., Xu J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg
In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems.We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such as the nonlinear Poisson-Boltzmann equation and its regularizations. The algorithm we study consists of the standard SOLVE-ESTIMATE-MARK-REFINE procedure common to many adaptive finite element algorithms, but where the SOLVE step involves only a full solve on the coarsest level, and the remaining levels involve only single Newton updates to the previous approximate solution.