AbstractThe paper is concerned with the reliable numerical solution of a class of nonlinear interface problems
governed by the Poisson–Boltzmann equation. Arising in electrostatic biomolecular models these problems
typically contain measure-type source terms and their solution often exposes drastically different behaviour
in different subdomains. The interface conditions reflect the requirement that the potential and its normal
derivative must be continuous.
In the first part of the paper, we discuss an appropriate weak formulation of the problem that guarantees existence
and uniqueness of the generalized solution. In the context of the considered class of nonlinear equations, this
question is not trivial and requires additional analysis, which is based on a special splitting of the problem into
simpler subproblems whose weak solutions can be defined in standard Sobolev spaces.
This splitting also suggests a rational numerical solution strategy and a way of deriving fully guaranteed error
bounds. These bounds (error majorants) are derived for each subproblem separately and, finally, yield a fully
computable majorant of the difference between the exact solution of the original problem and any energy-type
approximation of it.The efficiency of the suggested computational method is verified in a series of numerical tests related to real-life
biophysical systems.