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.

翻译：泊松-玻尔兹曼方程是一类带有狄拉克分布源的非线性椭圆型方程，已被广泛应用于溶液中生物分子系统静电势的预测。本文讨论并分析了一般多面体网格上泊松-玻尔兹曼方程的虚拟元方法。在全局解的低正则性条件下，我们获得了虚拟元近似在L2范数和H1范数下接近最优的误差估计。在不同多面体网格上的数值实验表明，虚拟元方法具有高效性，并验证了所提出的理论预测。