Virtual Element Methods For Poisson-boltzmann Equations

Poisson-Boltzmann equation(PBE)is widely used in the fields of biology,chemistry and physics.The virtual element method is a new discrete method for solving partial differential equations in recent years,which is suitable for arbitrary polygon and arbitrary polyhedron meshes.Compared with the traditional discrete methods such as finite element and finite difference,the virtual element method has lower requirements for mesh and enables wide adaptability.Because of the strong nonlinearity and singularity of PBE,it is difficult to analyze the theory and implement the numerical algorithm.In this paper,the virtual element discrete schemes are designed for PBEs,the error analysis under the H1-and L2-norm of the virtual element solution is given,and the virtual element calculation of PBE in the polygon meshes is realized.It is mainly divided into the following two parts.Firstly,for a class of linearized PBE(linear PBE for short),two different discrete forms of virtual elements are designed,which are respectively based on a elliptic projection operator and an L2 projection operator.The H1 norm error analysis of numerical solutions in two discrete forms is given.Numerical experiments show the effectiveness of these two discrete forms and verify the results of theoretical analysis.Secondly,for a class of nonlinear PBE,a discrete form of virtual element based on an L2 projection operator is constructed,the convergence analysis under the H1-and L2-norm of virtual element solution is given,and the virtual element numerical calculation of PBE under polygonal meshes is realized.Numerical experiments verify the correctness of the theoretical analysis and show that the virtual element method is effective in solving PBE under polygonal meshes.