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

Authors:
Burak Aksoylu;Stephen D. Bond;Eric C. Cyr;Michael Holst
Affiliations:
Department of Mathematics, TOBB University of Economics and Technology, Ankara, Turkey 06560 and Department of Mathematics, Louisiana State University, Baton Rouge, USA 70803;Multiphysics Simulation Technologies Department, Sandia National Laboratories, Albuquerque, USA 87185;Scalable Algorithms Department, Sandia National Laboratories, Albuquerque, USA 87185;Department of Mathematics, University of California San Diego, La Jolla, USA 92093
Venue:
Journal of Scientific Computing
Year:
2012

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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.