| The typical Poisson-Boltzmann equation(PBE)is a type of nonlinear elliptic interface problem.The equation is a commonly used tool for describing the electrostatic potential of biological macromolecules in aqueous solutions and has wide applications in engineering and scientific fields.Its numerical solution has many difficulties,such as the extreme irregularity of the interface,multipole singularity,discontinuity of the discontinuous coefficient,and exponentially strong nonlinearity.The virtual element method(VEM),with its good mesh adaptability and flexibility,is one of the hotspots in the research of numerical methods for partial differential equations in recent years.This paper mainly studies the theoretical analysis and computation of a class of three-dimensional PBE VEMs.To this end,we conduct research from the following three aspects:Firstly,a virtual element discretization scheme on polyhedral meshes is constructed for a class of linear elliptic interface problems,and error estimates for the virtual element solution in L2-norm and H1-norm are provided.Theoretical analysis proves that under the conditions of a smooth interface and globally low regularity solutions,the L2-norm and H1norm errors of the virtual element solution can achieve a nearly optimal order.Numerical experiments are conducted to validate the correctness of the theoretical results.Secondly,based on the study of the above interface problem,an analysis and computation are given for a class of linearized PBE using VEM.Specifically,by decomposing and removing singularity,the original problem is transformed into a non-singular regularized PBE.Then,error analyses in L2-norm and H1-norm using VEM are provided,and the nearly optimal error estimates are obtained,which are order O(h2|logh|)and order O(h |log h |1/2),respectively.Numerical experiments using related three-dimensional VEMs are also conducted.Lastly,building upon the study of the linearized PBE,a VEM is presented for a typical class of nonlinear PBE.Through careful treatment of the nonlinear term(utilizing its differentiability and boundedness),nearly optimal error estimates in L2-norm and H1-norm for the virtual element solution are obtained,which are order O(h2 |log h|)and order O(h|logh|1/2),respectively.Relevant programs for three-dimensional VEMs on general polyhedral meshes are developed.Numerical results demonstrate the effectiveness of the VEM for solving the above model and validate the reliability of the proposed virtual element theoretical results. |
