We present an expanded mixed nite element approximation of second order elliptic problems containing a tensor coeecient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its ux (the tensor coeecient times the negative gradient). The resulting linear system is a saddle point problem. In the c...

We give an overview of superconvergence phenomena in the finite element method for solving three-dimensional problems, in particular, for elliptic boundary value problems of second order over uniform meshes. Some difficulties with superconvergence on tetrahedral meshes are presented as well. For a given positive integer m we prove that there is no tetrahedralization of R3 whose all edges are m-...

<p style='text-indent:20px;'>A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than optimal-order velocity of approximation proved both an energy norm and <inline-formula><tex-math id="M1">$ L^2 $</tex-math></inline-formula> norm. Op...

In this paper, we present a new approach to obtain the asymptotic expansion and superconvergence for the linear element on Union Jack mesh. First, we construct a generalized interpolation function and its discrete harmonic extension by using the energy embedding method and the method of separation of variables. Then, we present elaborate estimates for the generalized interpolation function and ...

In this paper, we discuss the superconvergence of the local discontinuous Galerkin methods for nonlinear convection-diffusion equations. We prove that the numerical solution is [Formula: see text]th-order superconvergent to a particular projection of the exact solution, when the upwind flux and the alternating fluxes are used. The proof is valid for arbitrary nonuniform regular meshes and for p...

The concept of the M-decomposition was introduced by Cockburn et al. (2017) to provide criteria guarantee optimal convergence rates for Hybridizable Discontinuous Galerkin (HDG) method coercive elliptic problems. In that paper they systematically constructed superconvergent hybridizable discontinuous methods approximate solutions PDEs on unstructured meshes. this paper, we use construct HDG Max...

Gradient recovery has been widely used for a posteriori error estimates (see Ainsworth & Oden, 2000; Babuška & Strouboulis, 2001; Chen & Xu, 2007; Fierro & Veeser, 2006; Zhang, 2007; Zienkiewicz et al., 2005; Zienkiewicz & Zhu, 1987, 1992a,b). Recently, it has been employed to enhance the eigenvalue approximations by the finite-element method under certain mesh conditions (see Naga et al., 2006...