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

TL;DR

This paper investigates adaptive finite element methods for nonlinear elliptic interface problems, specifically the Poisson-Boltzmann equation, using inexact solvers to improve computational efficiency while maintaining convergence.

Contribution

It introduces an AFEM algorithm with inexact solvers that reduces computational cost and proves its convergence, supported by numerical experiments.

Findings

The inexact AFEM converges with proven contraction properties.

The method maintains linear-time computational complexity.

Numerical results confirm the theoretical predictions.

Abstract

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. We summarize a recently developed AFEM convergence theory for inexact solvers, and present a sequence of numerical experiments that give evidence that the theory does in fact predict the contraction properties of AFEM with inexact solvers. The various routines used are all designed to maintain a linear-time computational complexity.