Adaptive Finite Element Methods with Inexact Solvers for the Nonlinear Poisson-Boltzmann Equation
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.
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