Corrigendum to “G2 B-spline interpolation to a closed mesh” [Comput Aided Des 43 (2011) 145–160]

A $G^2$ interpolation method to a closed mesh was proposed by Shi et al. in [1]. This note points out that their method has two theoretical defects, which may lead to the failure of $G^2$ continuity. We fix these defects and correct several formula mistakes.     

A novel approach of high accuracy analysis of anisotropic bilinear finite element for time-fractional diffusion equations with variable coefficient

In this paper, by using bilinear finite element and $L1$ approximation, a fully-discrete scheme is established for time fractional diffusion equation with variable coefficient on anisotropic meshes. Unconditionally stable analysis of the proposed scheme are presented in both $L^2$-norm and $H^1$-norm. Moreover, convergence, superclose and superconvergence results are derived by combining interpolation with projection, which is the key technique for the numerical analysis. Specifically, by defining a novel projection operator, the error estimate between the projection and the exact solution is obtained on anisotropic meshes. Furthermore, high accuracy analysis on interpolation of bilinear finite element and projection is gained by means of some known results about the interpolation and mean value technique. Based on the related results about projection and interpolation, the optimal error estimate in $L^2$-norm and superclose of interpolation in $H^1$-norm are deduced by skillfully dealing with fractional derivative. At the same time, the global superconvergence is presented by employing interpolation postprocessing operator. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.     

H2-stability of the first order Galerkin method for the Boussinesq equations with smooth and non-smooth initial data

In this paper, the $H^2$-stability of the first order fully discrete Galerkin finite element methods for the Boussinesq equations with smooth and non-smooth initial data is presented. The finite element spatial discretization for the Boussinesq equations is based on the mixed finite element method, and the temporal treatments of the spatial discrete Boussinesq equations include the implicit scheme, the semi-implicit scheme, the implicit/explicit scheme and the explicit scheme. The $H^2$-stability results of the above numerical schemes are established. Firstly, we prove that the implicit and semi-implicit schemes are the $H^2$-unconditional stable. Then we show that the implicit/explicit scheme is $H^2$-almost unconditional stable with the initial data that belong to $H^1$ and $H^2$, and the similar results are obtained for the semi-implicit/explicit scheme in the case of the initial data that belong to $L^2$. Furthermore, we show that the explicit scheme is the $H^2$-conditional stable. Finally, some numerical examples are provided to verify the established theoretical findings and confirm the corresponding $H^2$ stability analysis of the different numerical schemes.     

Unconditional convergence and optimal L2 error estimates of the Crank–Nicolson extrapolation FEM for the nonstationary Navier–Stokes equations

In this paper, we study stability and convergence of fully discrete finite element method on large timestep which used Crank–Nicolson extrapolation scheme for the nonstationary Navier–Stokes equations. This approach bases on a finite element approximation for the space discretization and the Crank–Nicolson extrapolation scheme for the time discretization. It reduces nonlinear equations to linear equations, thus can greatly increase the computational efficiency. We prove that this method is unconditionally stable and unconditionally convergent. Moreover, taking the negative norm technique, we derive the $L^2$, $H^1$-unconditionally optimal error estimates for the velocity, and the $L^2$-unconditionally optimal error estimate for the pressure. Also, numerical simulations on unconditional$L^2$-stability and convergent rates of this method are shown.