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In this work, an effective and fast finite element numerical method with high‐order accuracy is discussed for solving a nonlinear time fractional diffusion equation. A two‐level linearized finite element scheme is constructed and a temporal–spatial error splitting argument is established to split the error into two parts, that is, the temporal error and the spatial error. Based on the regularity of the time discrete system, the temporal error estimate is derived. Using the property of the Ritz projection operator, the spatial error is deduced. Unconditional superclose result in H1‐norm is obtained, with no additional regularity assumption about the exact solution of the problem considered. Then the global superconvergence error estimate is obtained through the interpolated postprocessing technique. In order to reduce storage and computation time, a fast finite element method evaluation scheme for solving the nonlinear time fractional diffusion equation is developed. To confirm the theoretical error analysis, some numerical results are provided. [ABSTRACT FROM AUTHOR]

A finite difference/spectral method is proposed for the numerical approximation of a distributed order time fractional diffusion equation with initial singularity on two dimensional spatial domain. The L1 scheme on graded mesh for the discretization of time Caputo fractional derivative is used to capture the weak initial singularity and Legendre spectral method is adopted for the spatial discretization. It is proved that with appropriate choice of the grading parameter, the scheme can attain order 2 − β convergence in temporal direction, where β (0 < β < 1) is the upper bound of the order of distributed order fractional derivative. Numerical results confirm the sharpness of the error analysis. [ABSTRACT FROM AUTHOR]

In this study, we present several numerical approximation methods for solving the fractional diffusion equations with the Riesz space fractional derivative. Based on the fractional centered difference approximation to the Riesz fractional derivative, an explicit and an implicit finite difference schemes for solving the space fractional diffusion equations are obtained. The stability and global convergence of these two schemes in the maximum norm are analyzed rigorously. The convergence order is O ( τ + h 2 ) , where τ is the temporal grid size and h is spatial grid size, respectively. Furthermore, a numerical scheme for the one-dimensional multi-term time–space fractional diffusion equations is obtained. The unconditionally stability and convergence of the scheme in maximum norm are established. The convergence orders are 2 - α in the temporal direction and two in the spatial direction, where α is the maximum time fractional derivative orders in the equation. In addition, the extension to the two-dimensional case is also discussed. Finally, several numerical examples are provided to show the effectiveness and accuracy of our methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a least squares Galerkin–Petrov nonconforming mixed finite element method (LSGPNMFM) is proposed and analyzed for the stationary Conduction–Convection problem. We use -nonconforming as approximation space for the velocity, the linear element for the pressure space and the quadratic element for the temperature space. The mixed finite element spaces and need not satisfy inf–sup condition, the existence, uniqueness and convergence of the discrete solution are presented and error estimates of optimal order are derived in the case of sufficient viscosity. [Copyright &y& Elsevier]

Abstract: The main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary Navier–Stokes equations under anisotropic meshes. That is, the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure. The convergence analysis is presented and the optimal error estimates both in a broken -norm for the velocity and in an -norm for the pressure are derived on anisotropic meshes. [Copyright &y& Elsevier]

Abstract: In this paper, a new stable nonconforming mixed finite element scheme with second order accuracy is proposed for the stationary Stokes and Navier–Stokes equations, in which, a new nonconforming rectangular element is taken for approximating space the velocity and the bilinear element for the pressure. The optimal error estimates for the approximation of both the velocity and the pressure in -norm are established, as well as one in a broken -norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this paper, the discretization of the stationary Stokes and Navier–Stokes equations in a two-dimensional domain by a nonconforming mixed finite element method is considered. The standard formulation of the Stokes and Navier–Stokes equations in the primitive variables is formed, which takes for approximating space the new nonconforming rectangular element for the velocity and the bilinear element for the pressure. The optimal error estimates for the approximations of both the velocity in broken -norm and the pressure in -norm are achieved. Numerical experiments are given which are consistent with our theoretical analysis. [Copyright &y& Elsevier]