Binjie Li

Binjie Li, Hao Luo, Xiaoping Xie

This paper establishes the convergence of a time-steeping scheme for time fractional diffusion problems with nonsmooth data. We first analyze the regularity of the model problem with nonsmooth data, and then prove that the time-steeping scheme possesses optimal convergence rates in $ L^2(0,T;L^2(Ω)) $-norm and $ L^2(0,T;H_0^1(Ω)) $-norm with respect to the regularity of the solution. Finally, numerical results are provided to verify the theoretical results.

Binjie Li, Xiaoping Xie

In this paper, we analyze a family of hybridizable discontinuous Galerkin (HDG) methods for second order elliptic problems in two and three dimensions. The methods use piecewise polynomials of degree $k\geqslant 0$ for both the flux and numerical trace, and piecewise polynomials of degree $k+1 $ for the potential. We establish error estimates for the numerical flux and potential under the minimal regularity condition. Moreover, we construct a local postprocessing for the flux, which produces a numerical flux with better conservation. Numerical experiments in two-space dimensions confirm our theoretical results.

Binjie Li, Hao Luo, Xiaoping Xie

This paper analyzes a space-time finite element method for fractional wave problems. The method uses a Petrov-Galerkin type time-stepping scheme to discretize the time fractional derivative of order $ γ$ ($1<γ<2$). We establish the stability of this method, and derive the optimal convergence in the $ H^1(0,T;L^2(Ω)) $-norm and suboptimal convergence in the discrete $ L^\infty(0,T;H_0^1(Ω)) $-norm. Furthermore, we discuss the performance of this method in the case that the solution has singularity at $ t= 0 $, and show that optimal convergence rate with respect to the $ H^1(0,T;L^2(Ω)) $-norm can still be achieved by using graded grids in the time discretization. Finally, numerical experiments are performed to verify the theoretical results.

Binjie Li, Xiaoping Xie, Shiquan Zhang

This paper analyzes an abstract two-level algorithm for hybridizable discontinuous Galerkin (HDG) methods in a unified fashion. We use an extended version of the Xu-Zikatanov (X-Z) identity to derive a sharp estimate of the convergence rate of the algorithm, and show that the theoretical results also apply to weak Galerkin (WG) methods. The main features of our analysis are twofold: one is that we only need the minimal regularity of the model problem; the other is that we do not require the triangulations to be quasi-uniform. Numerical experiments are provided to confirm the theoretical results.

Binjie Li, Hao Luo, Xiaoping Xie

This paper analyzes a time-stepping discontinuous Galerkin method for modified anomalous subdiffusion problems with two time fractional derivatives of orders $ α$ and $ β$ ($ 0 < α< β< 1 $). The stability of this method is established, the temporal accuracy of $ O(τ^{m+1-β/2}) $ is derived, where $m$ denotes the degree of polynomials for the temporal discretization. It is shown that, even the solution has singularity near $ t = {0+} $, this temporal accuracy can still be achieved by using the graded temporal grids. Numerical experiments are performed to verify the theoretical results.

Binjie Li, Hao Luo, Xiaoping Xie

This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence of this algorithm are derived, and numerical experiments are performed, demonstrating the exponential decay in the temporal discretization error provided the solution is sufficiently smooth.

Binjie Li, Xiaoping Xie

This paper proposes and analyzes an optimal preconditioner for a general linear symmetric positive definite (SPD) system by following the basic idea of the well-known BPX framework. The SPD system arises from a large number of nonstandard finite element methods for diffusion problems, including the well-known hybridized Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) mixed element methods, the hybridized discontinuous Galerkin (HDG) method, the Weak Galerkin (WG) method, and the nonconforming Crouzeix-Raviart (CR) element method. We prove that the presented preconditioner is optimal, in the sense that the condition number of the preconditioned system is independent of the mesh size. Numerical experiments are provided to confirm the theoretical results.

Binjie Li, Xiaoping Xie

This paper analyzes a two-level algorithm for the weak Galerkin (WG) finite element methods based on local Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) mixed elements for two- and three-dimensional diffusion problems with Dirichlet condition. We first show the condition numbers of the stiffness matrices arising from the WG methods are of $O(h^{-2})$. We use an extended version of the Xu-Zikatanov (XZ) identity to derive the convergence of the algorithm without any regularity assumption. Finally we provide some numerical results.