Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation

TL;DR

This paper introduces goal-oriented error indicators and a problem-specific marking strategy for adaptive refinement in solving the Poisson-Boltzmann equation, demonstrating improved convergence for solvation free energy calculations.

Contribution

It develops a novel marking strategy based on solvation free energy and integrates multilevel preconditioning to enhance adaptive mesh refinement efficiency.

Findings

Goal-oriented indicators alone are insufficient for optimal refinement.

The proposed marking strategy improves convergence of solvation free energy.

Multilevel preconditioning maintains optimal computational complexity.

Abstract

In this article, we develop goal-oriented error indicators to drive adaptive refinement algorithms for the Poisson-Boltzmann equation. Empirical results for the solvation free energy linear functional demonstrate that goal-oriented indicators are not sufficient on their own to lead to a superior refinement algorithm. To remedy this, we propose a problem-specific marking strategy using the solvation free energy computed from the solution of the linear regularized Poisson-Boltzmann equation. The convergence of the solvation free energy using this marking strategy, combined with goal-oriented refinement, compares favorably to adaptive methods using an energy-based error indicator. Due to the use of adaptive mesh refinement, it is critical to use multilevel preconditioning in order to maintain optimal computational complexity. We use variants of the classical multigrid method, which can be viewed as generalizations of the hierarchical basis multigrid and Bramble-Pasciak-Xu (BPX) preconditioners.