Adaptive Goal-Oriented Solver for the Linearized Poisson- Boltzmann Equation

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. The new solver is based on a derivation of a new goal-oriented a posteriori error estimate for the electrostatic coupling. This estimate involves the consideration of the primal and adjoint problems for the electrostatic potential of the system, where the goal functional requires pointwise evaluations of the potential. A common practice to treat such functionals is to regularize them, for example, by averaging over balls or by mollification, and to use weak formulations in standard Sobolev spaces. However, this procedure changes the goal functional and may require the numerical evaluation of integrals of discontinuous functions. We overcome the conceptual shortcomings of this approach by using weak formulations involving nonstandard Sobolev spaces and deriving a representation of the error in the goal quantity which does not require averaging and directly exploits the original goal functional. The accuracy of this solver is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths. All the numerical experiments are repeated by using the well-known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver.