Reliable numerical solution of a class of nonlinear elliptic problems generated by the Poisson-Boltzmann equation

TL;DR

This paper develops guaranteed, fully computable error bounds for numerical solutions of nonlinear elliptic problems modeled by the Poisson-Boltzmann equation, with theoretical proofs and numerical validation in 2D and 3D.

Contribution

It introduces a rigorous error estimation framework for PBE-related problems, including error identities, majorants, and minorants, enabling reliable and adaptive numerical solutions.

Findings

Proved mathematical correctness of the nonlinear elliptic problems.

Derived guaranteed bounds for approximation errors.

Validated the approach with numerical tests in 2D and 3D domains.

Abstract

We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson-Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp., 69:481-500, 2000] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computable majorants and minorants of the global error as well as indicators of local errors that provide efficient adaptation of meshes. Theoretical results are confirmed by a collection of numerical tests that includes problems on $2D$ and $3D$ Lipschitz domains.