Search

The recovered gradient, using the polynomial preserving recovery (PPR), is constructed for the finite volume element method (FVEM) under simplex meshes. Regarding the main results of this paper, there are two aspects. Firstly, we investigate the supercloseness property of the FVEM, specifically examining the quadratic FVEM under tetrahedral meshes. Secondly, we present several guidelines for selecting computing nodes such that the least-squares fitting procedure of the PPR admits a unique solution. Numerical experiments demonstrate that the recovered gradient by the PPR exhibits superconvergence. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the superconvergence properties in approximation of the second- and third-order BVPs by spectral collocation methods. The theoretical analyses identify the superconvergence points of the interpolation function. Ample numerical experiments are carried out which perfectly match the theoretical results. In addition, numerical results show that the superconvergence properties also hold in solving PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we present and study C1 Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree k (≥ 3) for one-dimensional elliptic equations. We prove that, the solution and its derivative approximations converge with rate 2k − 2 at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree k + 1 in each element, the first-order derivative approximation is superconvergent at all interior k − 2 Lobatto points, and the second-order derivative approximation is superconvergent at k − 1 Gauss points, with an order of k + 2, k + 1, and k, respectively. As a by-product, we prove that both the Petrov-Galerkin solution and the Gauss collocation solution are superconvergent towards a particular Jacobi projection of the exact solution in H2, H1, and L2 norms. All theoretical findings are confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study the theory of convergence and superconvergence for integer and fractional derivatives of the one-point and two-point Hermite interpolations. When considering the integer-order derivatives, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates are O(N−2) and O(N−1.5) better than the global rates for the one-point and two-point interpolations, respectively. Here N represents the degree of the interpolation polynomial. It is proved that the αth fractional derivative of (u−uN), with k<α

The contribution of this article contains two parts: first, we prove a supercloseness result for the partially penalized immersed finite element (PPIFE) methods in (Lin, T. Lin, Y. & Zhang, X. (2015), Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer. Anal. 53, 1121–1144) and then based on the supercloseness result, we show that the gradient recovery method proposed in our previous work (Guo, H. & Yang, X. (2017) Gradient recovery for elliptic interface problem: II. Immersed finite element methods. J. Comput. Phys. 338, 606–619) can be applied to the PPIFE methods and the recovered gradient converges to the exact gradient with a superconvergent rate $$\mathcal{\rm O}(h^{1.5})$$. Hence, the gradient recovery method provides an asymptotically exact a posteriori error estimator for the PPIFE methods. Several numerical examples are presented to verify our theoretical results. [ABSTRACT FROM AUTHOR]

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]

In this paper, we study superconvergence properties of the discontinuous Galerkin method using upwind-biased numerical fluxes for one-dimensional linear hyperbolic equations. A (2k + 1)th order superconvergence rate of the DG approximation at the numerical fluxes and for the cell average is obtained under quasi-uniform meshes and some suitable initial discretization, when piecewise polynomials of degree k are used. Furthermore, surprisingly, we find that the derivative and function value approximation of the DG solution are superconvergent at a class of special points, with an order k +1 and k+2, respectively. These superconvergent points can be regarded as the generalized Radau points. All theoretical findings are confirmed by numerical experiments. [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 paper, we propose two systematic strategies to recover the gradient on the boundary of a domain. The recovered gradient has comparable superconvergent property on the boundary as that in the interior of the domain. This superconvergence property has been validated by several numerical experiments. [ABSTRACT FROM AUTHOR]

A polynomial preserving recovery method is introduced for over-penalized symmetric interior penalty discontinuous Galerkin solutions to a quasi-linear elliptic problem. As a post-processing method, the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns, as well as general meshes satisfying Condition ( ϵ , σ ) . By means of the averaging technique, we prove the polynomial preserving recovery method for averaged solutions is superconvergent, satisfying similar estimates as those for conforming finite element methods. We deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact. Extensive numerical results consistent with our analysis are presented. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we present and analyze a -version finite volume method (FVM) for elliptic problems in the one dimensional setting. Under some regularity assumptions of the exact solution, it is shown that the -version FV solution converges with exponential rates under and -norms. Superconvergence properties at nodal, Lobatto and Gauss points have been also discussed. Numerical results are presented to support our theoretical findings. [Copyright &y& Elsevier]

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]