Multigrid Algorithms Of Adaptive Finite Volume Element Method For Second Order Elliptic Problem

Finite volume element method,also called covolume method,box method and generalized difference method,is one of the important method to solve the partial differential equations,which is based on finite difference method and finite element method.The finite volume element method includes not only the merit of computational simplicity of finite difference method,but also the high precision as good as finite element method.Specially,finite volume element method keeps the local integral conservation,which is a significant property used in fluid mechanics and other subject areas.In this paper,we study the multigrid V-cycle method of finite volume element method on adaptively refined meshes for second-order elliptic problems using the newest vertex bisection algorithms,which is carried on the meshes where a posteriori error is larger than others.This may reduce computation and improve calculation accuracy.We treat the finite volume element method as a perturbation of the finite element method.With the help of the convergence result of the multigrid algorithms of adaptive finite element method,we derive the uniform convergence of the multigrid V-cycle method of adaptive finite volume element method by establishing two assumed conditions.The results of numerical experiments confirm our theoretical findings.