A unifying theory of a posteriori finite element error control

Summary Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm |||| of a linear functional of the formin the variable of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker of . As a consequence, any residual estimator that is a computable bound of |||| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.Supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.     

A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results.     

A unifying theory of a posteriori error control for discontinuous Galerkin FEM

A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617–637, 2005) and Carstensen and Hu (J Numer Math 107(3):473–502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied to the Laplace, Stokes, and Lamé equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature. The estimate recovers known results as well as provides new bounds to a number of schemes. C. Carstensen and M. Jensen supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the Hausdorff Institute of Mathematics in Bonn, Germany. C. Carstensen, T. Gudi, and M. Jensen supported by DST-DAAD (PPP-05) project no. 32307481.     

A posteriori error estimates for nonconforming finite element methods

Summary. Computable a posteriori error bounds for a large class of nonconforming finite element methods are provided for a model Poisson-problem in two and three space dimensions. Besides a refined residual-based a posteriori error estimate, an averaging estimator is established and an -estimate is included. The a posteriori error estimates are reliable and efficient; the proof of reliability relies on a Helmholtz decomposition. Received March 4, 1997 / Revised version received September 4, 2001 / Published online December 18, 2001     

A new a posteriori error estimate for the Morley element

In this paper, we present a posteriori error analysis for the nonconforming Morley element of the fourth order elliptic equation. We propose a new residual-based a posteriori error estimator and prove its reliability and efficiency. These results refine those of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) by dropping two edge jump terms in both the energy norm of the error and the estimator, and those of Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) by showing the efficiency in the sense of Verfürth (A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, New York, 1996). Moreover, the normal component in the estimators of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) and Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) is dropped, and therefore only the tangential component of the stress on each edge comes into the estimator. In addition, we generalize these results to three dimensional case.     

A posteriori estimation of the linearization error for strongly monotone nonlinear operators

We investigate the a posteriori estimation of the modeling (or linearization) error which arises when a nonlinear problem is replaced by a linear model. Using the context of strongly monotone operators, we construct a computable upper estimator for this error, and also provide an estimator that gives a lower bound. Several numerical results illustrating our theory are provided.     

Pointwise a posteriori error estimates for monotone semi-linear equations

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature.     

Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications

In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented.     

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 dual-mixed adaptive finite element error control for Lamé and Stokes equations

A unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the Herrmann formulation within the Hellinger-Reissner principle. This quasi-optimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) is called Dual-Mixed Hybrid formulation (DMH). Explicit residual-based a posteriori error estimates for DMH are introduced and are mathematically shown to be locking-free, reliable, and efficient. The estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation. Numerical evidence supports that the adaptive scheme leads to optimal convergence for Lamé and Stokes benchmark problems with singularities.     

A posteriori error estimators for the first-order least-squares finite element method

In this paper, we propose a posteriori error estimators for certain quantities of interest for a first-order least-squares finite element method. In particular, we propose an a posteriori error estimator for when one is interested in where . Our a posteriori error estimators are obtained by assigning proper weight (in terms of local mesh size h_T) to the terms of the least-squares functional. An a posteriori error analysis yields reliable and efficient estimates based on residuals. Numerical examples are presented to show the effectivity of our error estimators.     

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.     

A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients

We perform the a posteriori error analysis of residual type of transmission problem with sign changing coefficients. According to Bonnet-BenDhia et al. (2010) [9], if the contrast is large enough, the continuous problem can be transformed into a coercive one. We further show that a similar property holds for the discrete problem for any regular meshes, extending the framework from Bonnet-BenDhia et al. [9]. The reliability and efficiency of the proposed estimator are confirmed by some numerical tests.     

An a priori error estimate for a monotone mixed finite-element discretization of a convection–diffusion problem

We present a local exponential fitting hybridized mixed finite-element method for convection–diffusion problem on a bounded domain with mixed Dirichlet Neuman boundary conditions. With a new technique that interpretes the algebraic system after static condensation as a bilinear form acting on certain lifting operators we prove an a priori error estimate on the Lagrange multipliers that requires minimal regularity. While an extension of more classical arguments provide an estimate for the other solution components.     

Besov regularity and new error estimates for finite volume approximations of the p-laplacian

Summary. In [1], we have constructed a family of finite volume schemes on rectangular meshes for the p-laplacian and we proved error estimates in case the exact solution lies in W^2,p. Actually, W^2,p is not a natural space for solutions of the p-laplacian in the case p>2. Indeed, for general L^p data it can be shown that the solution only belongs to the Besov space In this paper, we prove Besov kind a priori estimates on the approximate solution for any data in L^p. We then obtain new error estimates for such solutions in the case of uniform meshes     

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

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

A posteriori error estimators for convection-diffusion equations

Summary. We derive a posteriori error estimators for convection-diffusion equations with dominant convection. The estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh-Peclet number. The estimators are either based on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems. Received February 10, 1997 / Revised version received November 4, 1997     

Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes

Summary. Both for the - and -norms, we prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes. We also show that, with a correct scaling, edge residuals yield a robust error estimator for a singularly perturbed reaction-diffusion equation. Received April 19, 1999 / Published online April 20, 2000     

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

Summary Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP ^th -degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 90-0194; by the U.S. Army Research Office under Contract Number DAAL03-91-G-0215; and by the National Science Foundation under Institutional Infrastructure Grant Number CDA-8805910     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε-weighted H¹-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.