Posteriori Error Control Research Articles

Consistent FE-analysis of Strang's problem based on mixed formulations

In this paper our studies on techniques for a posteriori error control and adaptive mesh design for finite element models in perfect plasticity are continued. The focus is on the numerical analysis of a low-order, dual-mixed discretisation. A posteriori error estimates are provided. Numerical tests confirm the theoretical results.

A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.

Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration**Dedicated to Ulrich Tautenhahn, a friend and co-author, who passed away too early at the age of 60.

The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which corrects both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration it is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.

A posteriori error estimator competition for conforming obstacle problems

AbstractThis 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

Pointwise a Posteriori Error Control for Discontinuous Galerkin Methods for Elliptic Problems

An a posteriori error bound for the maximum (pointwise) error for the interior penalty discontinuous Galerkin method for a standard elliptic model problem on polyhedral domains is presented. The computational domain is not required to be Lipschitz, thus allowing for domains with cracks and other irregular polyhedral domains. The proof is based on the direct use of Green's functions and varies substantially from the approach used in previous proofs of similar $L_\infty$ estimates for (continuous) finite element methods in the literature. Numerical experiments indicating the good behavior of the resulting a posteriori bounds within an adaptive algorithm are also presented.

A Priori and a Posteriori Error Analysis of the Discontinuous Galerkin Methods for Reissner-Mindlin Plates

AbstractIn this paper, we apply an a posteriori error control theory that we develop in a very recent paper to three families of the discontinuous Galerkin methods for the Reissner-Mindlin plate problem. We derive robust a posteriori error estimators for them and prove their reliability and efficiency.

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork bifurcation occurs when the underlying physical system possesses rotational and reflectional or O(2) symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual Weighted Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented. Here, particular attention is devoted to the problem of flow through a cylindrical pipe with a sudden expansion, which represents a notoriously difficult computational problem.

Adaptivity and a Posteriori Error Control for Bifurcation Problems II: Incompressible Fluid Flow in Open Systems with Z 2 Symmetry

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier–Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork or Hopf bifurcation occurs when the underlying physical system possesses reflectional or Z 2 symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual–Weighted–Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

Adaptive modeling of turbulent flow with residual based turbulent kinetic energy dissipation

In this paper we first review our recent work on a new framework for adaptive turbulence simulation: we model turbulence by weak solutions to the Navier–Stokes equations that are wellposed with respect to mean value output in the form of functionals, and we use an adaptive finite element method to compute approximations with a posteriori error control based on the error in the functional output. We then derive a local energy estimate for a particular finite element method, which we connect to related work on dissipative weak Euler solutions with kinetic energy dissipation due to lack of local smoothness of the weak solutions. The ideas are illustrated by numerical results, where we observe a law of finite dissipation with respect to a decreasing mesh size.

A FE/BE coupling for the 3D time-dependent eddy current problem. Part I: a priori error estimates

In this paper the discontinuous Galerkin method in time for the coupling of conforming finite element and boundary element methods is established. We derive quasi-optimal a priori error estimates. Numerical examples prove the new scheme to be useful in practice. A posteriori error control and an adaptive algorithm are studied in Part II of this paper.

A Posteriori Error Control for Discontinuous Galerkin Methods for Parabolic Problems

We derive energy-norm a posteriori error bounds for an Euler time-stepping method combined with various spatial discontinuous Galerkin schemes for linear parabolic problems. For accessibility, we address first the spatially semidiscrete case and then move to the fully discrete scheme by introducing the implicit Euler time-stepping. All results are presented in an abstract setting and then illustrated with particular applications. This enables the error bounds to hold for a variety of discontinuous Galerkin methods, provided that energy-norm a posteriori error bounds for the corresponding elliptic problem are available. To use this method in practice, we apply it to the interior penalty discontinuous Galerkin method, for which new a posteriori error bounds are derived. For the analysis of the time-dependent problems we use the elliptic reconstruction technique, and we deal with the nonconforming part of the error by deriving appropriate computable a posteriori bounds for it. We illustrate the theory with a series of numerical experiments aimed at (1) exploring practically the reliability and efficiency of the derived a posteriori estimates, and (2) testing them in an adaptive algorithm implementation.

A Posteriori Error Analysis of Finite Element Methods for Reissner–Mindlin Plates

This paper develops an a posteriori error control theory of finite element methods for Reissner–Mindlin plates, which states that one can derive a uniformly reliable and efficient a posteriori error estimate for a given scheme by only: (1) checking three conditions; (2) designing three functions and one parameter; (3) bounding the last three terms of the abstract estimator. We apply this theory to two classes of methods and achieve robust a posteriori error controls for them.

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 Lame 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.

Error control in h- and hp-adaptive FEM for Signorini's problem

This paper presents a posteriori finite element error estimates for Signorini’s problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The a posteriori error control relies on estimating the discretization error of an auxiliary problem which is given as a variational equation. The estimation consists of error bounds for the discretization error of the auxiliary problem and some further terms which capture the geometrical error and the error in the complementary condition. The derived estimates are applied to hand hp-adaptive refinement and enrichment strategies. Numerical results confirm the applicability of the theoretical findings. In particular, optimal algebraic and almost exponential convergence rates are obtained.

Unified framework for an a posteriori error analysis of non-standard finite element approximations of H(curl)-elliptic problems

Recently, a unified framework for a residual-based a posteriori error analysis of standard conforming finite element methods as well as non-standard techniques such as nonconforming and mixed methods has been developed. This paper provides such a framework for an a posteriori error control of nonconforming finite element discretizations of H(curl)-elliptic problems as they arise from low-frequency electromagnetics. These nonconforming approximations include, among others, the interior penalty discontinuous Galerkin (IPDG) approach.

New adaptive mixed finite element method (AMFEM)

AbstractThe need to develop reliable and efficient adaptive algorithms using mixed finite element methods arises from various applications in fluid dynamics and computational continuum mechanics. In order to save degrees of freedom, not all but just some selected set of finite element domains are refined and hence the fundamental question of convergence requires a new mathematical argument as well as the question of optimality.We will present a new adaptive algorithm for mixed finite element methods to solve the model Poisson problem, for which optimal convergence can be proved. The a posteriori error control of mixed finite element methods dates back to Alonso (1996) Error estimators for a mixed method. and Carstensen (1997) A posteriori error estimate for the mixed finite element method. The error reduction and convergence for adaptive mixed finite element methods has already been proven by Carstensen and Hoppe (2006) Error Reduction and Convergence for an Adaptive Mixed Finite Element Method, Convergence analysis of an adaptive nonconforming finite element methods.Recently, Chen, Holst and Xu (2008) Convergence and Optimality of Adaptive Mixed Finite Element Methods. presented convergence and optimality for adaptive mixed finite element methods following arguments of Rob Stevenson for the conforming finite element method. Their algorithm reduces oscillations, before applying and a standard adaptive algorithm based on usual error estimation. The proposed algorithm does this in a natural way, by switching between the reduction of either the estimated error or oscillations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Variational Principle for Hardening Elastoplasticity

We present a variational principle governing the quasi-static evolution of a linearized elastoplastic material. In the case of linear hardening, the novel characterization allows us to recover and partly extend some known results and proves itself to be especially well suited for discussing general approximation and convergence issues. In particular, the variational principle is exploited in order to prove in a novel setting the convergence of time and space-time discretizations as well as to provide some possible a posteriori error control.

A posteriori error control of a state constrained elliptic control problem

A posteriori error control of a state constrained elliptic control problem

New a posteriori error indicator in terms of linear functionals for linear elliptic problems

This paper is concerned with a posteriori error estimates in terms of goal-oriented quantities for linear elliptic problems. The quantities are presented by locally supported linear functionals that serve as measures of the difference between the exact solution and its approximation in subdomains that can be selected using certain a priori knowledge on the expected properties of the solution. Typically, the evaluation of such type of error indicators is performed with the help of the so-called adjoint problem (see, e.g., [W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Berlin, 2003; R. Becker and R. Rannacher, Weighted a posteriori error control in finite element methods. In: Proc. of ENUMATH'97, 1998, pp. 621–637]). In [S. Korotov, P. Neittaanmäki, and S. Repin, A posteriori error estimation of goal-oriented quantities by the superconvergent patch recovery. J. Numer. Math. (2003) 11, No. 1, 33–59] an approach was suggested, in which the primal and adjoint problems are solved on non-coinciding meshes, and averaging of gradients is used to evaluate the term in the estimator that cannot be computed directly. However, superconvergence requires higher differentiability of the solutions and also regularity (quasi-uniformity) of both primal and dual meshes. In the present paper we propose a new method for computation of the estimator, where we use only superconvergence of the post-processed finite-element solution of the primal problem. At this point our analysis is based on the properties of the least square surface fitting operator investigated by J. Wang [J. Wang, A Superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems. J. Math. Study (2000) 33, No. 3, 229–243]. We establish asymptotical properties of the new error indicator and verify its efficiency in a series of numerical tests.

A review ofa posteriorierror control and adaptivity for approximations of non‐linear conservation laws

AbstractIn this contribution we give an overview on recent progress in obtaininga posteriorierror control for finite volume and discontinuous Galerkin approximations of non‐linear hyperbolic conservation laws. The theory is based on the celebrated doubling of variables technique introduced by Kružkov (Math. USSR Sb. 1970;10:217–243).A posteriorierror control is of particular importance as it can be used for designing efficient grid adaptive schemes. The derivation of such adaptive methods is discussed and numerical experiments are given. Copyright © 2007 John Wiley & Sons, Ltd.