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.     

A posteriori error estimators for mixed finite element methods in linear elasticity

Summary. Three a posteriori error estimators for PEERS and BDMS elements in linear elasticity are presented: one residual error estimator and two estimators based on the solution of auxiliary local problems with different boundary conditions. All of them are reliable and efficient with respect to the standard norm and furthermore robust for nearly incompressible materials.Correspondence to: R. Verfürth     

Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms

Two residual-based a posteriori error estimators of the nonconforming Crouzeix-Raviart element are derived for elliptic problems with Dirac delta source terms.One estimator is shown to be reliable and efficient,which yields global upper and lower bounds for the error in piecewise W1,p seminorm.The other one is proved to give a global upper bound of the error in Lp-norm.By taking the two estimators as refinement indicators,adaptive algorithms are suggested,which are experimentally shown to attain optimal convergence orders.     

A posteriori error estimates for mixed FEM in elasticity

A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from inverse estimates. The constants in both estimates are independent of the Lamé constant , and so locking phenomena for are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement. Received July 17, 1997     

Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions

The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true error as the resources of the used computer allow. This work was supported by the Academy Research Fellowship No. 208628 from the Academy of Finland.     

A posteriori error estimates for mixed finite element approximations of elliptic problems

We derive residual based a posteriori error estimates of the flux in L ²-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.     

A posteriori error estimates for mixed finite element approximations of parabolic problems

We derive residual based a posteriori error estimates for parabolic problems on mixed form solved using Raviart–Thomas–Nedelec finite elements in space and backward Euler in time. The error norm considered is the flux part of the energy, i.e. weighted L ²(Ω) norm integrated over time. In order to get an optimal order bound, an elementwise computable post-processed approximation of the scalar variable needs to be used. This is a common technique used for elliptic problems. The final bound consists of terms, capturing the spatial discretization error and the time discretization error and can be used to drive an adaptive algorithm.     

Residual-based a posteriori error estimate for a mixed Reißner-Mindlin plate finite element method

Reliable and efficient residual-based a posteriori error estimates are established for the stabilised locking-free finite element methods for the Reissner-Mindlin plate model. The error is estimated by a computable error estimator from above and below up to multiplicative constants that do neither depend on the mesh-size nor on the plate's thickness and are uniform for a wide range of stabilisation parameter. The error is controlled in norms that are known to converge to zero in a quasi-optimal way. An adaptive algorithm is suggested and run for improving the convergence rates in three numerical examples for thicknesses 0.1, .001 and .001.     

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.     

Energy norm a posteriori error estimates for mixed finite element methods

This paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations. It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method. The analysis is performed in two different ways: under a saturation assumption and using a Helmholtz decomposition for vector fields.

     

A posteriori error estimates of mixed DG finite element methods for linear parabolic equations

In this article, we analyse a posteriori error estimates of mixed finite element discretizations for linear parabolic equations. The space discretization is done using the order λ?≥?1 Raviart–Thomas mixed finite elements, whereas the time discretization is based on discontinuous Galerkin (DG) methods (r?≥?1). Using the duality argument, we derive a posteriori l ^∞(L ²) error estimates for the scalar function, assuming that only the underlying mesh is static.     

A posteriori error estimates for mixed finite element solutions of convex optimal control problems

In this paper, we present an a posteriori error analysis for mixed finite element approximation of convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates can be used to construct reliable adaptive mixed finite elements for the control problems.     

Superconvergence and a posteriori error estimation for triangular mixed finite elements

Summary. In this paper,we prove superconvergence results for the vector variable when lowest order triangular mixed finite elements of Raviart-Thomas type [17] on uniform triangulations are used, i.e., that the -distance between the approximate solution and a suitable projection of the real solution is of higher order than the -error. We prove results for both Dirichlet and Neumann boundary conditions. Recently, Duran [9] proved similar results for rectangular mixed finite elements, and superconvergence along the Gauss-lines for rectangular mixed finite elements was considered by Douglas, Ewing, Lazarov and Wang in [11], [8] and [18]. The triangular case however needs some extra effort. Using the superconvergence results, a simple postprocessing of the approximate solution will give an asymptotically exact a posteriori error estimator for the -error in the approximation of the vector variable. Received December 6, 1992 / Revised version received October 2, 1993     

Residual-based a posteriori error control for Space-time finite element approximations of parabolic optimal control problems

In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed,where the stress is approximated by symmetric H(div)-Pk polynomial tensors and the displacement is approximated by C-1-Pk-1polynomial vectors,for all k 4.The main ingredients for the analysis are a new basis of the space of symmetric matrices,an intrinsic H(div)bubble function space on each element,and a new technique for establishing the discrete inf-sup condition.In particular,they enable us to prove that the divergence space of the H(div)bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued Pk-1polynomial space on each tetrahedron.The optimal error estimate is proved,verified by numerical examples.     

New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017     

A posteriori error estimator competition for conforming obstacle problems

This article on the a posteriori error analysis of the obstacle problem with affine obstacles and Courant finite elements compares five classes of error estimates for accurate guaranteed error control. To treat interesting computational benchmarks, the first part extends the Braess methodology from 2005 of the resulting a posteriori error control to mixed inhomogeneous boundary conditions. The resulting guaranteed global upper bound involves some auxiliary partial differential equation and leads to four contributions with explicit constants. Their efficiency is examined affirmatively for five benchmark examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.     

A family of mixed finite elements for linear elasticity

Summary A family of finite elements for use in mixed formulations of linear elasticity is developed. The stresses are not required to be symmetric, but only to satisfy a weaker condition based upon Lagrange multipliers. This is based on the same formulation used in the PEERS finite element spaces. Elements for both two and three dimensional problems are given. Error analysis on these elements is done, and some superconvergence results are proved.