Abstract:
In this paper, we present a new regularization scheme for the linearized Poisson–
Boltzmann equation (PBE) which models the electrostatic potential of biomolecules in a solvent.
This scheme is based on the splitting of the target potential into the short- and long-range compo-
nents localized in the molecular region by using the range-separated (RS) tensor format [P. Benner,
V. Khoromskaia, and B. N. Khoromskij, SIAM J. Comput., 2 (2018), pp. A1034–A1062] for represen-
tation of the discretized multiparticle Dirac delta [B. N. Khoromskij, J. Comput. Phys., 401 (2020),
108998] constituting the highly singular right-hand side in the PBE. From the computational point of
view our regularization approach requires only the modification of the right-hand side in the PBE so
that it can be implemented within any open-source grid-based software package for solving PBE that
already includes some FEM/FDM disretization scheme for elliptic PDE and solver for the arising lin-
ear system of equations. The main computational benefits are twofold. First, one applies the chosen
PBE solver only for the smooth long-range (regularized) part of the collective potential with the regu-
lar right-hand side represented by a low-rank RS tensor with a controllable precision. Thus, we elimi-
nate the numerical treatment of the singularities in the right-hand side and do not change the interface
and boundary conditions. And second, the elliptic PDE need not be solved for the singular part in the
right-hand side at all, since the short-range part of the target potential of the biomolecule is precom-
puted independently on a computational grid by simple one-dimensional tensor operations. The total
potential is then obtained by adding the numerical solution of the PBE for the smooth long-range part
to the directly precomputed tensor representation for the short-range contribution. Numerical tests
illustrate that the new regularization scheme, implemented by a simple modification of the right-hand
side in the chosen PBE solver, improves the accuracy of the approximate solution on rather coarse
grids. The scheme also demonstrates good convergence behavior on a sequence of refined grids.
