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Carsten Carstensen - One of the best experts on this subject based on the ideXlab platform.

  • residual based a Posteriori Error analysis for symmetric mixed arnold winther fem
    Numerische Mathematik, 2019
    Co-Authors: Carsten Carstensen, Dietmar Gallistl, Joscha Gedicke
    Abstract:

    This paper introduces an explicit residual-based a Posteriori Error analysis for the symmetric mixed finite element method in linear elasticity after Arnold–Winther with pointwise symmetric and \(H({\text {div}})\)-conforming stress approximation. The residual-based a Posteriori Error estimator of this paper is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof. Hence, it is straightforward to implement an adaptive mesh-refining algorithm. Numerical experiments verify the proven reliability and efficiency of the new a Posteriori Error estimator and illustrate the improved convergence rate in comparison to uniform mesh-refining. A higher convergence rate for piecewise affine data is observed in the \(L^2\) stress Error and reproduced in non-smooth situations by the adaptive mesh-refining strategy.

  • residual based a Posteriori Error analysis for symmetric mixed arnold winther fem
    arXiv: Numerical Analysis, 2017
    Co-Authors: Carsten Carstensen, Dietmar Gallistl, Joscha Gedicke
    Abstract:

    This paper introduces an explicit residual-based a Posteriori Error analysis for the symmetric mixed finite element method in linear elasticity after Arnold-Winther with pointwise symmetric and H(div)-conforming stress approximation. Opposed to a previous publication, the residual-based a Posteriori Error estimator of this paper is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof. Hence it is straightforward to implement an adaptive mesh-refining algorithm obligatory in practical computations. Numerical experiments verify the proven reliability and efficiency of the new a Posteriori Error estimator and illustrate the improved convergence rate in comparison to uniform mesh-refining. A higher convergence rates for piecewise affine data is observed in the L2 stress Error and reproduced in non-smooth situations by the adaptive mesh-refining strategy.

  • a Posteriori Error control for dpg methods
    SIAM Journal on Numerical Analysis, 2014
    Co-Authors: Carsten Carstensen, Leszek Demkowicz, Jayadeep Gopalakrishnan
    Abstract:

    A combination of ideas in least-squares finite element methods with those of hybridized methods recently led to discontinuous Petrov--Galerkin (DPG) finite element methods. They minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard, locally computable, dual norm. This paper establishes a general a Posteriori Error analysis for the natural norms of the DPG schemes under conditions equivalent to a priori stability estimates. It is proven that the locally computable residual norm of any discrete function is a lower and an upper Error bound up to explicit data approximation Errors. The presented abstract framework for a Posteriori Error analysis applies to known DPG discretizations of Laplace and Lame equations and to a novel DPG method for the stress-velocity formulation of Stokes flow with symmetric stress approximations. Since the Error control does not rely on the discrete equations, it applies to inexactly computed or otherwise perturbed solutions within the discrete space...

  • a unifying theory of a Posteriori Error control for nonconforming finite element methods
    Numerische Mathematik, 2007
    Co-Authors: Carsten Carstensen
    Abstract:

    Residual-based a Posteriori Error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in Carstensen [Numer Math 100:617–637, 2005]. Therein, the key assumption is that the conforming first-order finite element space $$V_h^c$$ annulates the linear and bounded residual l written $$V_h^c \subseteq {\rm ker} \ell$$. That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that $$V_h^c \not\subset {\rm ker} \ell$$. The present paper generalises the aforementioned theory to more general situations to deduce new a Posteriori Error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator $$\Pi: V_h^c \rightarrow V_h^{nc}$$ with some elementary properties. It is conjectured that the more general hypothesis (H1)–(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and Navier–Lame equations illustrate the presented unifying theory of a Posteriori Error control for NCFEM.

  • quasi interpolation and a Posteriori Error analysis in finite element methods
    Mathematical Modelling and Numerical Analysis, 1999
    Co-Authors: Carsten Carstensen
    Abstract:

    One of the main tools in the proof of residual-based a Posteriori Error estimates is a quasi- interpolation operator due to Cl ement. We modify this operator in the setting of a partition of unity with the eect that the approximation Error has a local average zero. This results in a new residual- based a Posteriori Error estimate with a volume contribution which is smaller than in the standard estimate. For an elliptic model problem, we discuss applications to conforming, nonconforming and mixed nite element methods.

Ningning Yan - One of the best experts on this subject based on the ideXlab platform.

Dominik Schotzau - One of the best experts on this subject based on the ideXlab platform.

Rolf Rannacher - One of the best experts on this subject based on the ideXlab platform.

  • a Posteriori Error control for finite element approximations of elliptic eigenvalue problems
    Advances in Computational Mathematics, 2001
    Co-Authors: Vincent Heuveline, Rolf Rannacher
    Abstract:

    We develop a new approach to a Posteriori Error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a Posteriori Error representations are available with explicitly given remainder terms. The careful evaluation of these Error representations for the concrete situation of an eigenvalue problem results in a Posteriori Error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local Error indicators that are used in the mesh refinement process.

  • a Posteriori Error control in finite element methods via duality techniques application to perfect plasticity
    Computational Mechanics, 1998
    Co-Authors: Rolf Rannacher, F T Suttmeier
    Abstract:

    In this paper a new technique for a Posteriori Error control and adaptive mesh design is presented for finite element models in perfect plasticity. The approach is based on weighted a Posteriori Error estimates derived by duality arguments as proposed in Becker and Rannacher (1996) and Rannacher and Suttmeier (1997) for linear problems. The conventional strategies for mesh refinement in finite element methods are mostly based on a Posteriori Error estimates for the global energy norm in terms of local residuals of the computed solution. These estimates reflect the approximation properties of the trial functions by local interpolation constants while the stability property of the continuous model enters through a global coercivity constant. However, meshes generated on the basis of such global Error estimates are not appropriate in computing local quantities as point values or contour integrals and in the case of nonlinear material behavior. More accurate and efficient Error estimation can be achieved by using suitable weights which can be obtained numerically in the course of the refinement process from the solutions of linearized dual problems. This feed-back approach is developed here for primal-mixed finite element models in linear-elastic perfect plasticity.

Paul Houston - One of the best experts on this subject based on the ideXlab platform.