| The Fokker-Planck equation (FPE) is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion[6,28,42]. Let α be the highest order of the Fokker-Planck equation. When α=2, the FPE generates the seconder-order ad-vection dispersion equation (ADE). But solutes that moves through aquifers do not generally follow the classic advection-dispersion equation because of large deviations from the stochastic process of Brownian motion, but it can be described by the fractional advection-dispersion equation [25,31,32,43].Because of the nonlocal property of fractional derivative or integral op-erators, numerical methods of fractional diffusion equations generate full or dense matrices. Traditional methods to solve these numerical methods re-quires computational cost of O(N3) per time step and O(N2) of memory, where N is the number of spatial grid points. The significant computational work and memory impose a serious challenge for numerical simulation of high dimensional space-fractional diffusion equations. Meerschaert and Tadjeran [29,30] utilized a shifted Grunwald-Letnikov finite difference approximation to develop an implicit finite difference method for (one-dimensional) space-fractional diffusion equations, and proved that the method is unconditional stable and has first-order convergence rate. Wang et al [53,54] developed a fast finite difference methods for fractional diffusion equations with Dirichlet boundary conditions. The stiffness matrix can be decomposed as a sum of diagonal-multiply-Toeplitz matrices. Consequently, they get a fast numeri-cal method which only requires computational cost of O(N log N) and O(N) of memory per Krylov subspace method iteration, while retaining the same accuracy and approximation property as regular finite difference methods. Because of the nonlocal nature of the fractional differential operator, the resulting numerical approximation to the Neumann boundary condition cou-ples the unknowns in the entire domain and destroys the structure of the stiffness matrix. Wang and Du [50] developed a superfast-preconditioned conjugate gradient squared method for the efficient solution of steady-state space-fractional diffusion equations with uniform mesh. The method reduces the computational work from O(N2) to O(N log N) per iteration and re-duces the memory requirement from O(N2) to O(N). It was realized recently [22,55,56] that solutions to fractional diffusion equations may exhibit bound-ary layer and poor regularity even if the diffusivity coefficient and right-hand side are smooth. From a numerical point of view, a numerical scheme with a uniform mesh is probably not anticipated to efficiently resolve the boundary layer of the solution u at x= 0, a locally refined composite mesh is desired. ● finite difference methods are nonconservative. ● finite difference methods have one-order accuracy. ● finite difference methods are defined on uniform meshes. ● finite volume methods are conservative. ● finite volume methods have two-order accuracy. ● finite volume methods can be applied for nonuniform meshes.In many applications, conservation property is crucial. Thus we use finite vol-ume methods to solve fractional differential equations with composite meshes. However, in the context of fractional differential equations, any local change in a mesh will destroy the global structure of the stiffness matrix. Hence, one should take great care in the development of an adaptive method by delicately balancing the adaptivity of the method and the structure of the adapted meshes.An optimal-order error estimate for Galerkin finite element methods for time-dependent convection-diffusion equations with non-degenerate diffusion were proved in [24,45,48,51,58] via the introduction of the Ritz projection. The advantage of this estimate is that the estimate is valid for any regular partition. However, because the approximation property of Ritz projection depends on the Peclet number of the problem, so the error estimate derived in [24,45,48,51,58] also depends on the Peclet number, and blows up as the lower bound of the diffusion approaches to zero.In this article, we developed a preconditioned fast Krylov subspace meth-ods, and give an uniform error estimate of degenerated convection diffusion equations of bilinear Galerkin finite element method and finite difference method. The rest of this article is organized as follows. In section I we give the basic introduction of the convection-diffusion equations, including the classic advection-diffusion equations and fractional advection-diffusion equations. In section II, we give some preliminaries. In section III, we de-veloped a preconditioned fast method of fractional diffusion equations with fractional derivatives boundary conditions, by exploring the structure of it-s stiffness matrix. In section IV, we developed a preconditioned fast finite volume method on locally refined composite meshes. In section V, we de-veloped a preconditioned fast finite volume method on triangular mesh for fractional diffusion equations. In section Ⅵ, we proved a uniformly optimal-order error estimate in a degenerate-diffusion weighted energy norm for bi-linear Galerkin finite element methods for two-dimensional time-dependent advection-diffusion equations with degenerate diffusion. In section VII, we prove a uniformly optimal-order error estimate for a finite difference method for degenerate convection-diffusion equations. |
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