| The finite volume element method is a commonly used discretization method for solving partial differential equations,its main advantage is that it maintains the local conservation of physical quantities.Therefore,the finite volume element method is widely applied in many fields such as fluid dynamics,petroleum engineering,reser-voir modeling,and so on.In this article,we mainly studies several problems related to the finite volume element method for convection-diffusion equations,which include the correction of preserving discrete extremum principle for low-order finite volume element methods,the scheme construction and theoretical analysis of the new upwind finite volume element method for convection-diffusion-reaction equations,and the the-oretical analysis of the new upwind finite volume element method for time-dependent convection-diffusion-reaction equations.In the first part,for diffusion equations,we study the extremum-preserving correc-tion of the low-order finite volume element method.Taking the diffusion equation as a model,we consider the linear element finite volume methods on triangular grids and the bilinear element finite volume method on quadrilateral grids.We apply the nonlin-ear correction technique recently proposed by scholars to the aforementioned low-order finite volume element methods,which ensures that the corrected scheme uncondition-ally satisfies the discrete extremum principle for the mesh partition.And the corrected scheme still maintains the conservation,i.e.,flux continuity.The main idea of the cor-rection is to decompose the numerical flux into the principal part with a two-point flux structure and the defective part.Then,using the local extremums,we correct the defec-tive part into the nonlinear numerical flux with a two-point flux form.It is proved that the corrected scheme unconditionally satisfies the discrete extremum principle.Further-more,we extend the extremum-preserving correction technique to the time-dependent problems.Numerous numerical experiments indicate that the corrected scheme not only satisfies the discrete extremum principle but also preserves the optimal convergence or-der of the original scheme.In the second part,for convection-diffusion-reaction problems,we construct and analyze a new upwind finite volume element scheme on general quadrilateral meshes.We discrete the diffusion term and reaction term by standard bilinear finite volume method.It is well-known that if the discretization of the convection term uses the tra-ditional upwind finite volume method,the scheme is stable and avoids non-physical oscillations,but its convergence rate in L~2norm drops to first order.In this paper,we define a new upwind scheme for the convection term by using two terms Taylor expan-sion.We prove the stability of the new upwind finite volume method,and provide the optimal order error estimates in H~1norm and L~2norm.A large number of numerical experiments confirm the correctness of the theoretical results,especially whether for dominant diffusion or dominant convection,that the convergence rate of the numerical scheme is optimal second order in L~2norm.In the third part,for convection-diffusion problems,we consider the extremum-preserving correction of the finite volume element method.The extremum-preserving correction for the diffusion term is the same as in the first part.For the discretization of the convection term,we use the new upwind scheme introduced in the second part,which is stable but does not satisfy the discrete extremum principle.We decompose the discretized numerical flux of the convection term into the principal part with standard upwind characteristic and the defective part.Different from the principal part of the dif-fusion term,the principal part of the convection term does not have two-point flux struc-ture.For the stiffness matrix corresponding to the principal part of the convection term,the diagonal elements are non-negative,the off-diagonal elements are non-positive,and each row has only one non-zero element.The correction method for the defective part of the numerical flux of the convection term is the same as that for the defective part of the diffusion term.We prove that the corrected scheme unconditionally satisfies the discrete extremum principle.Extensive numerical experiments demonstrate that the corrected scheme satisfies the discrete extremum principle while preserving the optimal convergence order of the original new upwind finite volume element scheme.In the fourth part,for convection-diffusion problems,we consider the new up-wind finite volume element method.For spatial discretization,we use the new upwind finite volume element method introduced in the second part,while for temporal dis-cretization,we utilize the backward Euler scheme.For the semi-discrete finite volume element scheme,we provide the stability analysis,and optimal order error estimates in H~1norm.For the fully discrete backward Euler finite volume element scheme,we prove the stability,and optimal order error estimates in H~1norm and L~2norm.Numeri-cal experiments confirm the correctness of the theoretical results,whether for dominant diffusion or dominant convection,the convergence rate of the backward Euler new up-wind finite volume element scheme in H~1norm is O(τ+h)and in L~2norm is O(τ+h~2),both of them reach the optimal order. |
