A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.

     

特征值问题协调/非协调有限元后验误差分析

本文研究对称椭圆特征值问题的有限元后验误差估计,包括协调元和非协调元,具有下列特色:(1)对协调/非协调元建立了有限元特征函数uh的误差与相应的边值问题有限元解的误差在局部能量模意义下的恒等关系式,该边值问题的右端为有限元特征值λh与uh的乘积,有限元解恰好为uh.从而边值问题有限元解在能量模意义下的局部后验误差指示子,包括残差型和重构型后验误差指示子,成为有限元特征函数在能量模意义下的局部后验误差指示子.(2)讨论了协调有限元特征函数的基于插值后处理的梯度重构型后验误差估计,对有限元特征函数的导数得到了最大模意义下的渐近准确局部后验误差指示子.     

A posteriori error analysis for nonconforming approximations of an anisotropic elliptic problem

We develop in this article an a posteriori error estimator for the P₁‐nonconforming finite element approximation, for a diffusion‐reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015     

A posteriori error estimation and adaptivity in the method of lines with mixed finite elements

We will investigate the possibility to use superconvergence results for the mixed finite element discretizations of some time-dependent partial differential equations in the construction of a posteriori error estimators. Since essentially the same approach can be followed in two space dimensions, we will, for simplicity, consider a model problem in one space dimension.     

A posteriori analysis of the finite element discretization of some parabolic equations

We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respect to both time and space approximations and proving their equivalence with the error, in order to work with adaptive time steps and finite element meshes.

RÉSUMÉ. Nous considérons la discrétisation d'équations paraboliques, soit linéaires soit semi-linéaires, par un schéma d'Euler implicite en temps et par éléments finis en espace. L'idée de cet article est de construire des indicateurs d'erreur liés à l'approximation en temps et en espace et de prouver leur équivalence avec l'erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d'éléments finis adaptés à la solution.

     

An augmented mixed finite element method with Lagrange multipliers: A priori and a posteriori error analyses

In this paper, we provide a priori and a posteriori error analyses of an augmented mixed finite element method with Lagrange multipliers applied to elliptic equations in divergence form with mixed boundary conditions. The augmented scheme is obtained by including the Galerkin least-squares terms arising from the constitutive and equilibrium equations. We use the classical Babuška–Brezzi theory to show that the resulting dual-mixed variational formulation and its Galerkin scheme defined with Raviart–Thomas spaces are well posed, and also to derive the corresponding a priori error estimates and rates of convergence. Then, we develop a reliable and efficient residual-based a posteriori error estimate and a reliable and quasi-efficient Ritz projection-based one, as well. Finally, several numerical results illustrating the performance of the augmented scheme and the associated adaptive algorithms are reported.     

A priori and a posteriori error analysis for the mixed discontinuous Galerkin finite element approximations of the biharmonic problems

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L² norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L² norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017     

A posteriori estimates for approximations of time-dependent Stokes equations

Charalambos Makridakis ^{In this paper, we derive a posteriori error estimates for space-discreteapproximations of the time-dependent Stokes equations. By usingan appropriate Stokes reconstruction operator, we are able towrite an auxiliary error equation, in pointwise form, that satisfiesthe exact divergence-free condition. Thus, standard energy estimatesfrom partial differential equation theory can be applied directly,and yield a posteriori estimates that rely on available correspondingestimates for the stationary Stokes equation. Estimates of optimalorder in L(L2) and L(H1) for the velocity are derived for finite-elementand finite-volume approximations.}     

A posteriori error estimate for finite volume element method of the parabolic equations

In this article, residual‐type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual‐type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual‐type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259–275, 2017     

A FE‐BE coupling for a fluid‐structure interaction problem: Hierarchical a posteriori error estimates

In this article, we establish a hierarchical a posteriori error estimate for a coupling of finite elements and boundary elements for a fluid‐structure interaction problem posed in two and three dimensions. These methods combine boundary elements for the exterior fluid and finite elements for the elastic structure. We consider two weak formulations, a nonsymmetric one and a symmetric one, which are both uniquely solvable. We present the reliability and efficiency of the error estimates. For the two dimensional case, we compute local error indicators which allow us to develop an adaptive mesh refinement strategy on triangles. For the three dimensional case, we use hexahedrons as elements. Numerical experiments underline our theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

A posteriori analysis of finite element discretizations of stochastic partial differential delay equations

In this paper, we study a posteriori error estimates for finite element approximation of stochastic partial differential delay equations containing a noise. We derive an energy norm a posteriori bounds for an Euler time-stepping method combined with a standard Galerkin schemes for the problems. For accessibility, we first address the spatially semidiscrete case and then move to the fully discrete scheme.     

Local a posteriori error estimates and adaptive control of pollution effects

Local a posteriori error estimators are derived for linear elliptic problems over general polygonal domains in 2d. The estimators lead to a sharp upper bound for the energy error in a local region of interest. This upper bound consists of H¹‐type local error indicators in a slightly larger subdomain, plus weighted L²‐type local error indicators outside this subdomain, which account for the pollution effects. This constitutes the basis of a local adaptive refinement procedure. Numerical experiments show a superior performance than the standard global procedure as well as the generation of locally quasi‐optimal meshes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 421–442, 2003     

Reliable a posteriori error control for nonconforming finite element approximation of Stokes flow

We derive computable a posteriori error estimates for the lowest order nonconforming Crouzeix-Raviart element applied to the approximation of incompressible Stokes flow. The estimator provides an explicit upper bound that is free of any unknown constants, provided that a reasonable lower bound for the inf-sup constant of the underlying problem is available. In addition, it is shown that the estimator provides an equivalent lower bound on the error up to a generic constant.

     

A posteriori error analysis of a quadratic finite volume method for nonlinear elliptic problems

In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.     

Residual‐based a posteriori error estimator for the mixed finite element approximation of the biharmonic equation

A reliable and efficient residual‐based a posteriori error estimator is derived for the Ciarlet‐Raviart mixed finite element method for the biharmonic equation on polygonal domains. The performance of the estimator is illustrated by numerical experiments. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation

It is well-known that on uniform meshes the piecewise linearconforming finite element solution of the Poisson equation approximatesthe interpolant to a higher order than the solution itself.In this paper, this type of superclose property is studied forthe canonical interpolant defined by the nodal functionals ofseveral non-conforming finite elements of lowest order. By givingexplicit examples we show that some non-conforming finite elementsdo not admit the superclose property. In particular, we discusstwo non-conforming finite elements which satisfy the supercloseproperty. Moreover, applying a postprocessing technique, wecan also state a superconvergence property for the discretizationerror of the postprocessed discrete solution to the solutionitself. Finally, we show that an extrapolation technique leadsto a further improvement of the accuracy of the finite elementsolution.     

A posteriori error estimates of stabilized finite volume method for the Stokes equations

In this work, the residual‐type posteriori error estimates of stabilized finite volume method are studied for the steady Stokes problem based on two local Gauss integrations. By using the residuals between the source term and numerical solutions, the computable global upper and local lower bounds for the errors of velocity in H¹ norm and pressure in L² norm are derived. Furthermore, a global upper bound of u ? u_h in L²‐norm is also derived. Finally, some numerical experiments are provided to verify the performances of the established error estimators. Copyright © 2015 John Wiley & Sons, Ltd.     

Local a priori/a posteriori error estimates of conforming finite elements approximation for Steklov eigenvalue problems

Based on the work of Xu and Zhou(2000),this paper makes a further discussion on conforming finite elements approximation for Steklov eigenvalue problems,and proves a local a priori error estimate and a new local a posteriori error estimate in ||·||1,Ω0 norm for conforming elements eigenfunction,which has not been studied in existing literatures.     

Robust residual‐ and recovery‐based a posteriori error estimators for interface problems with flux jumps

For elliptic interface problems with flux jumps, this article studies robust residual‐ and recovery‐based a posteriori error estimators for the conforming finite element approximation. The residual estimator is a natural extension of that developed in [Bernardi and Verfürth, Numer Math 85 (2000), 579–608; Petzoldt, Adv Comp Math 16 (2002), 47–75], and the recovery estimator is a nontrivial extension of our method developed in Cai and Zhang, SIAM J Numer Anal 47 (2009) 2132–2156. It is shown theoretically that reliability and efficiency bounds of these error estimators are independent of the jumps provided that the distribution of the coefficients is locally monotone. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28:476–491, 2012     

A posteriori error analysis of nonconforming finite volume elements for general second‐order elliptic PDEs

In this article, we study the a posteriori H¹ and L² error estimates for Crouzeix‐Raviart nonconforming finite volume element discretization of general second‐order elliptic problems in ?². The error estimators yield global upper and local lower bounds. Finally, numerical experiments are performed to illustrate the theoretical findings. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011