Error Analysis of Virtual Element Method for the Poisson-Boltzmann Equation

TL;DR

This paper analyzes the virtual element method for solving the nonlinear Poisson-Boltzmann equation, providing error estimates and numerical validation on polyhedral meshes relevant to biological electrostatics.

Contribution

It offers nearly optimal error estimates for the virtual element method applied to the Poisson-Boltzmann equation on general meshes, addressing low regularity solutions.

Findings

Nearly optimal error estimates in L2 and H1 norms.

Numerical experiments confirm theoretical predictions.

Method is efficient on polyhedral meshes.

Abstract

The Poisson-Boltzmann equation is a nonlinear elliptic equation with Dirac distribution sources, which has been widely applied to the prediction of electrostatics potential of biological biomolecular systems in solution. In this paper, we discuss and analysis the virtual element method for the Poisson-Boltzmann equation on general polyhedral meshes. Under the low regularity of the solution of the whole domain, nearly optimal error estimates in both L2-norm and H1-norm for the virtual element approximation are obtained. The numerical experiment on different polyhedral meshes shows the efficiency of the virtual element method and verifies the proposed theoretical prediction.