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In this article, we analyze convergence and supercloseness properties of a class of weak Galerkin (WG) finite element methods for solving second-order elliptic problems. It is shown that the WG solution is superclose to the Lagrange interpolant using Lobatto points. This supercloseness behavior is obtained through some newly designed stabilization terms. A postprocessing technique using polynomial preserving recovery (PPR) is introduced for the WG approximation. Superconvergence analysis is performed for the PPR recovered gradient. Numerical examples are provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for one-dimensional linear convection-diffusion equations. We prove, under some suitable choice of numerical fluxes and initial discretization, a 2 k-th and [ABSTRACT FROM AUTHOR]

In this article, we combine mixed finite element method, multiscale discretization, and Rayleigh quotient iteration to propose a new adaptive algorithm based on residual type a posterior error estimates for the Stokes eigenvalue problem. Both reliability and efficiency of the error indicator are proved. The efficiency of the algorithm is also investigated using Chen's Innovation Finite Element Method (iFEM) package. Numerical results are satisfying.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 31-53, 2015 [ABSTRACT FROM AUTHOR]

We consider conforming finite element approximation of fourth-order singularly perturbed problems of reaction diffusion type. We prove superconvergence of standard C1 finite element method of degree p on a modified Shishkin mesh. In particular, a superconvergence error bound of [ABSTRACT FROM AUTHOR]

Considering a two-dimensional singularly perturbed convection-diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O( N-1 ln N) in a DG-norm is established under the regularity assumptions, while the total number of mesh points is O( N2). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 [ABSTRACT FROM AUTHOR]