In this paper, we investigate the accuracy-enhancement for the discontinuous Galerkin (DG) method for solving one-dimensional nonlinear symmetric systems of hyperbolic conservation laws. For nonlinear equations, the divided difference estimate is an important tool that allows for superconvergence of the post-processed solutions in the local L2 norm. Therefore, we first prove that the L2 norm of...

We analyze the use of a five-point difference formula for the discretization of the third derivative operator on nonuniform grids. The formula was derived so as to coincide with the standard five-point formula on regular grids and to lead to skew-symmetric schemes. It is shown that, under periodic boundary conditions, the formula is supraconvergent in the sense that, in spite of being inconsist...

Polynomial preserving recovery (PPR) was first proposed and analyzed in Zhang and Naga in SIAM J Sci Comput 26(4):1192–1213, (2005), with intensive following applications on elliptic problems. In this paper, we generalize the study of PPR to high-frequency wave propagation. Specifically, we establish the supercloseness between finite element solution and its interpolation with explicit dependen...

In this paper, we uncover and study a new superconvergence property of a large class of finite element methods for one-dimensional convectiondiffusion problems. This class includes discontinuous Galerkin methods defined in terms of numerical traces, discontinuous Petrov–Galerkin methods and hybridized mixed methods. We prove that the so-called numerical traces of both variables superconverge at...

Abstract. We present and analyze a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the linearized Korteweg-de Vries (KdV) equation in one space dimension. These estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We extend the work of Hufford and Xing [J. Comput. Appl. Math., 255 (20...

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [14] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also disc...

For the nonconforming rotated Q1 element over a mildly distorted quadrilateral mesh, we propose a superconvergence property at the element center, the vertices and the midpoints of four edges. Numerics are presented to confirm this observation.

The authors consider the biquadratic finite volume element approximation for the Pois-son's equation on the rectangular domain Ω = (0, 1) 2. The primal mesh is performed using a ractangular partition. The control volumes are chosen in such a way that the vertices are stress points of the primal mesh. In order to solve the scheme more efficiently, the authors wrote the bi-quadratic finite volume...

We develop multiscale mortar mixed finite element discretizations for second order elliptic equations. The continuity of flux is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. The polynomial degree of the mortar and subdomain approximation spaces may differ; in fact, the mortar sp...