Journal of Scientific Computing

, Volume 52, Issue 1, pp 202–225 | Cite as

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

  • Burak Aksoylu
  • Stephen D. BondEmail author
  • Eric C. Cyr
  • Michael Holst


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.


Poisson-Boltzmann equation Adaptive finite element methods Multilevel preconditioning Goal-oriented a posteriori error estimation Solvation free energy Electrostatics 


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Copyright information

© Springer Science+Business Media (outside the USA) 2011

Authors and Affiliations

  • Burak Aksoylu
    • 1
    • 2
  • Stephen D. Bond
    • 3
    Email author
  • Eric C. Cyr
    • 4
  • Michael Holst
    • 5
  1. 1.Department of MathematicsTOBB University of Economics and TechnologyAnkaraTurkey
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  3. 3.Multiphysics Simulation Technologies DepartmentSandia National LaboratoriesAlbuquerqueUSA
  4. 4.Scalable Algorithms DepartmentSandia National LaboratoriesAlbuquerqueUSA
  5. 5.Department of MathematicsUniversity of California San DiegoLa JollaUSA

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