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

This paper proposes and analyzes a weak Galerkin (WG) finite element method with strong symmetric stresses for two- and three-dimensional linear elasticity problems on conforming or nonconforming polygon/polyhedral meshes. The WG method uses piecewise-polynomial approximations of degrees k ( ≥ 1 ${\geq 1}$ ) for the stress, k + 1 ${k+1}$ for the displacement, and k for the displacement trace on the inter-element boundaries. It is shown to be equivalent to a hybridizable discontinuous Galerkin (HDG) finite element scheme. We show that the WG methods are robust in the sense that the derived a priori error estimates are optimal and uniform with respect to the Lamé constant λ. Numerical experiments confirm the theoretical results.

A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem - Volume 7 Issue 3 - Xiaobo Zheng, Xiaoping Xie