The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems

We study the gradient superconvergence of bilinear finite volume element (FVE) solving the elliptic problems. First, a superclose weak estimate is established for the bilinear form of the FVE method. Then, we prove that the gradient approximation of the FVE solution has the superconvergence property: $\max_{P\in S}|(\nabla u-\overline{\nabla}u_h)(P)|=O(h^2)|\ln h|$, where $\overline{\nabla}u_h(P)$ denotes the average gradient on elements containing point $P$ and $S$ is the set of optimal stress points composed of the mesh points, the midpoints of edges and elements

Any order superconvergence finite volume schemes for 1D general elliptic equations

We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy and $L^2$ norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special case, the convergence rate can reach $h^{2r}$, where $r$ is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.Comment: 24 pages, 6 figure

Finite volume schemes of any order on rectangular meshes

In this paper, we analyze vertex-centered finite volume method (FVM) of any order for elliptic equations on rectangular meshes. The novelty is a unified proof of the inf-sup condition, based on which, we show that the FVM approximation converges to the exact solution with the optimal rate in the energy norm. Furthermore, we discuss superconvergence property of the FVM solution. With the help of this superconvergence result, we find that the FVM solution also converges to the exact solution with the optimal rate in the $L^2$-norm. Finally, we validate our theory with several numerical experiments.Comment: 15 page

Is 2k2k-Conjecture valid for finite volume methods?

This paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes. Our main result is to prove {\it 2k-conjecture}: at each vertex of the underlying rectangular mesh, the bi-$k$ degree finite volume solution approximates the exact solution with an order $O(h^{2k})$, where $h$ is the mesh size. As byproducts, superconvergence properties for finite volume discretization errors at Lobatto and Gauss points are also obtained. All theoretical findings are confirmed by numerical experiments

Parametric Polynomial Preserving Recovery on Manifolds

This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not require the tangent spaces of the exact manifolds, and they have been assumed for some significant gradient recovery methods in the literature. Another advantage of PPPR is that superconvergence is guaranteed without the symmetric condition which has been asked in the existing techniques. There is also numerical evidence that the superconvergence by PPPR is high curvature stable, which distinguishes itself from the others. As an application, we show its capability of constructing an asymptotically exact \textit{a posteriori} error estimator. Several numerical examples on two-dimensional surfaces are presented to support the theoretical results and comparisons with existing methods are documented

Superconvergence of the Gradient Approximation for Weak Galerkin Finite Element Methods on Nonuniform Rectangular Partitions

This article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by the weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of ${\cal O}(h^r)$, $1.5\leq r \leq 2$, for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is ${\cal O}(h)$ for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory

Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart-Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss-Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal $k$-th order convergence for the velocity and pressure in their natural norms, as well as $(k+1)$-st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order $(k+1)$ in the full $L^2$-norm. Numerical results illustrating the validity of our theoretical results are included

Asymptotically exact a posteriori error estimates of eigenvalues by the Crouzeix-Raviart element and enriched Crouzeix-Raviart element

Two asymptotically exact a posteriori error estimates are proposed for eigenvalues by the nonconforming Crouzeix--Raviart and enriched Crouzeix-- Raviart elements. The main challenge in the design of such error estimators comes from the nonconformity of the finite element spaces used. Such nonconformity causes two difficulties, the first one is the construction of high accuracy gradient recovery algorithms, the second one is a computable high accuracy approximation of a consistency error term. The first difficulty was solved for both nonconforming elements in a previous paper. Two methods are proposed to solve the second difficulty in the present paper. In particular, this allows the use of high accuracy gradient recovery techniques. Further, a post-processing algorithm is designed by utilizing asymptotically exact a posteriori error estimators to construct the weights of a combination of two approximate eigenvalues. This algorithm requires to solve only one eigenvalue problem and admits high accuracy eigenvalue approximations both theoretically and numerically.Comment: arXiv admin note: text overlap with arXiv:1802.0189

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order $0<\alpha<1$. For each time $t \in [0,T]$, the HDG approximations are taken to be piecewise polynomials of degree $k\ge0$ on the spatial domain~$\Omega$, the approximations to the exact solution $u$ in the $L_\infty\bigr(0,T;L_2(\Omega)\bigr)$-norm and to $\nabla u$ in the $L_\infty\bigr(0,T;{\bf L}_2(\Omega)\bigr)$-norm are proven to converge with the rate $h^{k+1}$ provided that $u$ is sufficiently regular, where $h$ is the maximum diameter of the elements of the mesh. Moreover, for $k\ge1$, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for $u$ converging with a rate $h^{k+2}$ (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed