The Study Of Positivity-preserving Finite Volume Schemes And Vertex-centered Finite Volume Schemes For Anisotropic Diffusion Problems

We first propose a vertex-centered linearity-preserving finite volume(FV)scheme and its related decoupled and positivity-preserving discrete duality finite volume(DDFV)scheme for anisotropic diffusion problems on polyhedral meshes.By using a geometric relationship of the three-dimensional meshes to construct the cell matrix,the vertex-centered linearity-preserving FV scheme is constructed on polyhedrons.Based on this,we design a decoupled and positivity-preserving DDFV scheme on polyhedral meshes——a scheme which contains both the vertex unknowns and the cell-centered unknowns.We utilize the vertex-centered scheme to obtain ver-tex unknowns,introduce a positive correction for them,and then substitute the resulting vertex values into the FV equations constructed by the nonlinear two-point flux approximations on the primary mesh to obtain the cell-centered unknowns.The local conservation is strictly preserved on the primary mesh while conditionally maintained on the dual mesh for the decoupled and positivity-preserving DDFV scheme.Different from the traditional DDFV schemes,here the FV equations on the dual mesh can be solved independently,hence the two sets of FV equations are decoupled.Unlike the most nonlinear positivity-preserving schemes,no nonlinear iteration is required for linear problems and a general nonlinear solver could be used for nonlinear prob-lems.In addition,by comparing the Newton method with the fixed-point iteration method and its Anderson acceleration,the advantages of the presented positivity-preserving DDFV scheme are demonstrated by the numerical experiments.Second,for the two-dimensional linear diffusion problems on polygonal meshes,we pro-vide an theoretical analysis for the decoupled and positivity-preserving DDFV schemes,includ-ing the positivity,well-posedness,stability and H1 error estimates.Under some weak geometric assumptions,the stability and H1 error estimates for vertex unknowns are obtained.Further-more,by assuming the coercivity of the cell-centered FV equations,an H1 error estimate for the cell-centered unknowns is obtained.Numerical experiments verify the rationality of the coercive assumption and the correctness of other theoretical analysis results.Third,we propose an unconditionally stable vertex-centered linearity-preserving nine-point scheme for diffusion problems on quadrilateral meshes.The traditional cell-centered nine-point schemes have the advantages of simple flux expressions and easy to code,etc.How-ever,it has limitations in accuracy,theoretical analysis and the design of interpolation method for auxiliary unknowns.Following the construction of cell-centered nine-point schemes,we construct the FV equations on the dual counterpart of the quadrilateral mesh.By introducing a special stabilization technique,we finally obtain a family of unconditionally stable vertex-centered linearity-preserving nine-point schemes.These schemes fit well to the anisotropic and discontinuous cases,and maintain the advantages of cell-centered nine-point schemes.More important is that the stability and optimal H1 error estimations for the linear case are strict-ly proved.Unlike the vertex-centered linearity-preserving scheme in the first part,the linear system derived from this scheme is generally asymmetric.At last,we study the relationship between the vertex-centered linearity-preserving FV scheme(VLPS)and the lowest-order virtual element method(VEM)on the star-shaped polyg-onal meshes for diffusion problems,and propose a unified positive and local conservative post-processing for these two methods.As a generalization of the finite element method,VEM has made remarkable achievements in the algorithm and analysis of simulating various problems re-cently while VLPS is currently only applied to the numerical simulation of diffusion problems.From an algebraic point of view,the global stiffness matrix of VLPS with a special stabilization term coincides with that of the lowest-order VEM while the load terms are generally different.The global stiffness matrices of the two methods can be split as the consistency parts and the stability parts,and the consistency parts are the same while the stability parts coincide under some assumptions.At the same time,as a by-product,we obtain a new stability term for VLPS.Finally,numerical experiments confirm our theoretical findings.