The Finite Volume Element Method For Convection-diffusion Problems

The finite volume element method,also called generalized difference method,covolume method and box scheme,is a very important and main numerical discretization method for solving partial differential equations(PDEs).Due to the local integral conservation laws which satisfying,the finite volume element method could maintain the local conservation of the physical quantities,such as the mass,momentum and energy.Therefore,the FVEM has been widely applied to the fields of fluid dynamics,electromagnetism,petroleum engineering and other fields.This work contains two parts,which are relatively independent.The first part of this work considers the optimal weighted upwind finite volume method(OWUFVM)on rectangular mesh for solving the stationary convection-diffusion problem in 2D.For model problem with constant coefficients,we give the theoretical analysis of the OWUFVM,and the theory results include the stability of the scheme and H1-norm error estimate,the optimal error estimate in L2 norm,and the discrete maximum principle,where the optimal L2-norm estimate is the most important contribution in this part.About the second part,we take the time-dependent convection-diffusion problem defined on the moving domain as the model problem,and we construct the finite volume element method in Arbitrary Lagrangian-Eulerian framework(ALE-FVEM)to solve the problem.The semi-discrete and full-discrete ALE-FVEM schemes are developed,and the stability associated with each scheme is proved,respectively.For the work of the first part,we study the bilinear-element optimal weighted upwind finite volume method on rectangular mesh for solving the convection-diffusion problem in2 D.The OWUFVM is first proposed by Liang and Zhao in 2006(See [64]),and which introduces a non-standard dual partition in the construction of the method.In fact,the determinant of the dual mesh is dependent of a local Peclet's number,and the local Peclet's number is related to the convection velocity,diffusion coefficient and the size of the primary mesh.The main goal of this part is to provide the theoretical analysis of the OWUFVM for solving the convection-diffusion in constant coefficient case.Since the dual partition is non-standard,we first prove the positive definiteness of the bilinear form corresponding to the diffusion term through an element-by-element analysis strategy,and further present the stability of the method based on the positive definiteness result.Then we give the optimal error estimate in L2 norm,which is the most important result in this part.We would like to point out that the Aubin-Nistche technique is usually applied to the proof of L2-norm error estimate,however it may be not proper for current non-standard dual mesh.Thus,the weak error estimate of the first type is proposed to analysis the optimal L2-norm convergence.In the proof of the weak error estimate,the Taylor's expansion technique and the special selection of the non-standard dual mesh are employed.Finally,under some constrains of the grid,we prove that the OWUFVM satisfies the discrete maximum principle,which guarantees the numerical solution not to produce spurious oscillations.The time-dependent convection-diffusion problem defined on the moving domain is considered in the second part of the work.To solve the problem,the finite volume element method in ALE framework(ALE-FVEM)is firstly developed for solving the model problem.With the ALE formulation,two ALE counterparts about the problem are given to overcome the difficult from the moving domain,denoted by the non-conservation and conservation ALE form,respectively.Based on the two forms,the discretization of finite volume element is considered.To establish the ALE-FVEM,an element-preserving discrete ALE mapping is used,which could map the primary triangle element and the dual element on the initial reference domain to the triangle and dual element on a current domain for any time.Meantime,with the help of the discrete ALE mapping,the linear-element trial space and the piecewise constant test space on initial reference domain could also be defined on the current domain,respectively.So we can further obtain the semi-discretization ALE-FVEM with respect to the non-conservation and conservation form.And the stability of the two semi-discrete schemes are provided.On the basis of the semi discretization,we obtain the fully discrete ALE-FVEM by using the implicit Euler(IE)method.The space-time discretization schemes include:(1).the fully discrete ALEFVEM corresponding to the non-conservation form,(2).the fully discrete ALE-FVEM corresponding to the conservation form,(3).the fully discrete ALE-FVEM with GCL(Geometry Conservation Laws)corresponding to the conservation form.Then we prove the three full discretization schemes are stable.In these analyses,many techniques forsolving parabolic problem with FVEM on fixed domain can be extended to the moving one,and this is important for completing the proof.