Superconvergence and a posteriori error estimators of recovery type are analyzed for the 4-node hybrid stress quadrilateral finite element method proposed by Pian and Sumihara (Int. J. Numer. Meth. Engrg., 1984, 20: 1685-1695) for linear elasticity problems. Uniform superconvergence of order $O(h^{1+\min\{α,1\}})$ with respect to the Lamé constant $λ$ is established for both the recovered gradients of the displacement vector and the stress tensor under a mesh assumption, where $α>0$ is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. A posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.
In two dimensions, we propose and analyze an a posteriori error estimator for the acoustic spectral problem based on the virtual element method in $\H(÷;Ω)$. Introducing an auxiliary unknown, we use the fact that the primal formulation of the acoustic problem is equivalent to a mixed formulation, in order to prove a superconvergence result, necessary to despise high order terms. Under the virtual element approach, we prove that our local indicator is reliable and globally efficient in the $Ł^2$-norm. We provide numerical results to assess the performance of the proposed error estimator.
Superconvergence analysis of linear FEM based on the polynomial preserving recovery and Richardson extrapolation for Helmholtz equation with high wave number
Beijing Computational Science Research Center（北京计算科学研究中心）；Department of Mathematics, Nanjing University（南京大学数学系）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
We present a priori and superconvergence error estimates of mixed finite element methods for the pseudostress-velocity formulation of the Oseen equation. In particular, we derive superconvergence estimates for the velocity and a priori error estimates under unstructured grids, and obtain superconvergence results for the pseudostress under certain structured grids. A variety of numerical experiments validate the theoretical results and illustrate the effectiveness of the superconvergent recovery-based adaptive mesh refinement. It is also numerically shown that the proposed postprocessing yields…
Residual-based a posteriori error estimation for the Maxwell's eigenvalue problem
Dipartimento di Matematica “F. Casorati” Università degli studi di Pavia（帕维亚大学）；DICATAM Università degli Studi di Brescia（布雷西亚大学）；CI 2 MA Departamento de Ingeniería Matemática Universidad de Concepción（康塞普西翁大学）
We present two new recovery-based a posteriori error estimates for the Hellan--Herrmann--Johnson method in Kirchhoff--Love plate theory. The first error estimator uses a postprocessed deflection and controls the $L^2$ moment error and the discrete $H^2$ deflection error. The second one controls the $L^2\times H^1$ total error and utilizes superconvergent postprocessed moment field and deflection. The effectiveness of the theoretical results is numerically validated in several experiments.
In this article, we aim to recover locally conservative and $H(div)$ conforming fluxes for the linear Cut Finite Element Solution with Nitsche's method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart-Thomas space is completely local and does not require to solve any mixed problem. The $L^2$-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we are able to prove the global reliability and local efficiency.…
Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems
Chongqing Jiaotong University（重庆交通大学）；Texas A&M International University（德州农工国际大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Wayne State University（韦恩州立大学）
Virtual element methods is a new promising finite element methods using general polygonal meshes. Its optimal a priori error estimates are well established in the literature. In this paper, we take a different viewpoint. We try to uncover the superconvergent property of the virtual element methods by doing some local post-processing only on the degrees of freedom. Using linear virtual element method as an example, we propose a universal recovery procedure to improve the accuracy of gradient approximation for numerical methods using general polygonal meshes. Its capability of serving as a poste…
This work presents superconvergence estimates of the nonconforming Rannacher--Turek element for second order elliptic equations on any cubical meshes in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$. In particular, a corrected numerical flux is shown to be superclose to the Raviart--Thomas interpolant of the exact flux. We then design a superconvergent recovery operator based on local weighted averaging. Combining the supercloseness and the recovery operator, we prove that the recovered flux superconverges to the exact flux. As a by-product, we obtain a superconvergent recovery estimate of the Crouzei…
A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods
Université de Lille 1（里尔第一大学）；INRIA Lille Nord Europe（法国国家信息与自动化研究所里尔北部分支）；Université de Valenciennes et du Hainaut Cambrésis（瓦朗谢纳及康布雷埃斯海诺特大学）
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 …
In this paper, we introduce a simple local flux recovery for $\mathcal{Q}_k$ finite element of a scalar coefficient diffusion equation on quadtree meshes, with no restriction on the irregularities of hanging nodes. The construction requires no specific ad hoc tweaking for hanging nodes on $l$-irregular ($l\geq 2$) meshes thanks to the adoption of virtual element families. The rectangular elements with hanging nodes are treated as polygons as in the flux recovery context. An efficient a posteriori error estimator is then constructed based on the recovered flux, and its reliability is proved und…
This article is a review on basic concepts and tools devoted to a posteriori error estimation for problems solved with the Finite Element Method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems, approximated by a conforming numerical discretization. The review mainly aims at presenting in a unified manner a large set of powerful verification methods, around the concept of equilibrium. Methods based on that concept provide error bounds that are fully computable and mathematically certified. We discuss recovery methods, residual methods, and duality-…
In this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and a Hessian recovery based linear (HRBL) FEM for second order elliptic equations in non-divergence form. The elliptic equation is casted into a symmetric non-divergence weak formulation, in which second order derivatives of the unknown function are involved. We use gradient and Hessian recovery operators to calculate the second order derivatives of linear finite element approximations. Although, thanks to low degrees of freedom (DOF) of linear elements, the implementation of the proposed schemes is …
In this work we examine a posteriori error control for post-processed approximations to elliptic boundary value problems. We introduce a class of post-processing operator that `tweaks' a wide variety of existing post-processing techniques to enable efficient and reliable a posteriori bounds to be proven. This ultimately results in optimal error control for all manner of reconstruction operators, including those that superconverge. We showcase our results by applying them to two classes of very popular reconstruction operators, the Smoothness-Increasing Accuracy-Enhancing filter and Superconver…
We present and analyze a weak Galerkin finite element method for solving the transport-reaction equation in $d$ space dimensions. This method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon/polyhedra. We derive the \textcolor[rgb]{0.00,0.00,1.00}{$L_2$-error estimate} of $O(h^{k+\frac{1}{2}})$-order for the discrete solution when the $k$th-order polynomials are used for $k\geq 0$. Moreover, for a special class of meshes, we also obtain the \textcolor[rgb]{0.00,0.00,1.00}{optimal error} estimate of $O(h^{k+1})$-order in th…
This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic PDE. This allows us to derive a priori as well as a posteriori error bounds for the recovered states wi…
The Hessian discretisation method (HDM) for fourth order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit-conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods, finite volume methods and methods based on gradient recovery operators. A generic error estimate has been established in $L^2$, $H^1$ and $H^2$-like norms in literature. In this paper, we establish improved $L^2$ and $H^1$ error estimates in the framework of HDM and illustrate it on va…
We propose a mesh adaptation procedure for Cartesian quadtree meshes, to discretize scalar advection-diffusion-reaction problems. The adaptation process is driven by a recovery-based a posteriori estimator for the $L^2(Ω)$-norm of the discretization error, based on suitable higher order approximations of both the solution and the associated gradient. In particular, a metric-based approach exploits the information furnished by the estimator to iteratively predict the new adapted mesh. The new mesh adaptation algorithm is successfully assessed on different configurations, and turns out to perfor…
Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second-order PDEs. The non-divergence form in these problems is natural. If the coefficients of the diffusion matrix are not differentiable, the problem can not be transformed into the more convenient variational form. We investigate tailored non-conforming finite element approximations of second-order PDEs in non-dive…
