ARGOS (Adaptive Refinement Goal-Oriented Solver)

ARGOS is an adaptive finite element solver for the Linearized Poisson-Boltzmann equation (LPBE) currently in a prototype version written in FreeFem, see  "Hecht, F. New development in FreeFem++. J. Numer. Math. 2012, 20, 251-266.". This version is intended to supplement the paper "ARGOS: An Adaptive Refinement Goal-Oriented Solver for the Linearized Poisson-Boltzmann Equation" written by S. Nakov, E. Sobakinskaya, T. Renger, J. Kraus and available at https://doi.org/10.1002/jcc.26716. This version is limited to the models described in the paper and contains all the necessary data to reproduce the stated results. A fully working and optimized version written in C++ is planned for the future.

ARGOS is specifically designed for the computation of a goal quantity, i.e., the electrostatic interaction between molecules. The electrostatic interaction E between molecules M1 and M2 is equal to the sum of q_i.phi(x_i), where phi is the potential generated by the charges of M1, q_i are the charges of M2, and phi(x_i) is the potential phi evaluated at the position x_i of the charge q_i. Here, the potential phi is governed by the linearized Poisson-Boltzman equation (LPBE), a linear elliptic partial differential equation (PDE) with a measure source term:

                                -div(eps grad(phi)) + k^2.phi = F,

where eps is the dielectric coefficient, allowed to have a jump discontinuity across the molecule-solvent interface, k^2 is a piecewise constant that depends on the ionic strength Is, and F is the charge density of M1, which is a linear combination of delta functions centered at the point charges of M1. If Is>0, then the so-called ion exclusion layer (IEL) is added. This is a region in which no ions can penetrate and which surrounds the low dielectric cavities that represent the molecules M1 and M2. For details on the precise mathematical formulation of the PBE see https://www.sciencedirect.com/science/article/pii/S0022247X22000798?via%3Dihub and https://doi.org/10.1515/cmam-2020-0022.

The adaptive mesh refinement in ARGOS is based on a so-called goal oriented error indicator, computed by using the current approximation phi_h of the electrostatic potential due to M1 on the current mesh T_h. Since the goal quantity is just one number, the electrostatic interaction E, the purpose of the error indicator is to detect the regions for which the error in phi_h contributes the most to the error in the interaction E_h = sum [q_i.phi_h(x_i)]. Those regions are then being refined and are not necessarily the regions, where the error in phi_h (measured in a certain norm) is the highest.