A Posteriori Dual-Mixed (Hybrid) Adaptive Finite Element ... Posteriori Dual-Mixed (Hybrid) Adaptive Finite Element Error Control for Lame ... we will deal with a dual-mixed formulation

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  • A Posteriori Dual-Mixed (Hybrid) Adaptive FiniteElement Error Control for Lame and StokesEquations

    A Posteriori Dual-Mixed (Hybrid) Finite Element Error Control

    Carsten Carstensen, Paola Causin

    , Riccardo Sacco

    Department of Mathematics, Humboldt-Universitat zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany. E-Mail: cc@math.hu-berlin.deINRIA Rocquencourt, B.P.105 Domaine de Voluceau, F-78153 Le Chesnay Cedex, Roc-quencourt, France. E-Mail: paola.causin@inria.frMOX - Modeling and Scientific Computing, Dipartimento di Matematica F. Brioschi,Politecnico di Milano, via Bonardi 9, 20133 Milano, Italy.E-Mail: riccardo.sacco@mate.polimi.it

    Abstract A unified and robust mathematical model for compressible andincompressible linear elasticity can be obtained by rephrasing the Her-rmann formulation within the Hellinger-Reissner principle. This quasi-op-timally converging extension of PEERS (Plane Elasticity Element with Re-duced Symmetry) is called Dual-Mixed Hybrid formulation (DMH). Ex-plicit residual-based a posteriori error estimates for DMH are introducedand are mathematically shown to be locking-free, reliable, and efficient.The estimator serves as a refinement indicator in an adaptive algorithm foreffective automatic mesh generation. Numerical evidence supports that theadaptive scheme leads to optimal convergence for Lame and Stokes bench-mark problems with singularities.

    1 Introduction and motivation

    It is well known that for nearly incompressible and incompressible materi-als, i.e. for a value of the Poisson ratio near or equal to 0.5, finite elementcomputations based on a standard displacement formulation fail due tothe onset of the locking phenomenon (see [11] for numerical evidence). Avalid alternative to locking-affected methods is represented by dual-mixedformulations, that provide mathematical models capable of treating under

  • 2 Carsten Carstensen et al.

    an unified framework both compressible and incompressible linear elas-ticity problems (see [3,18]). However, the quasi-optimal convergence rateof such methods can be unfavorably degraded, for example, by the pres-ence of singularities in the computational domain. In such an event, theconvergence performance can be improved by resorting to a robust mesh-refinining algorithm for an efficient automatic mesh-design. A list of con-tributions proposing and analyzing robust and effective adaptive finite el-ement methods in compressible, nearly incompressible and pure incom-pressible solid and fluid mechanics includes references [4,10,11,13,12,17,23,24].

    In the sequel, we will deal with a dual-mixed formulation obtained byrephrasing the Herrmann approach [20] within the Hellinger-Reissner prin-ciple. This quasi-optimally converging extension of PEERS (Plane Elastic-ity Element with Reduced Symmetry) is called Dual-Mixed Hybrid formu-lation (DMH) and in the case of isotropic materials reads: Given the linearfunctionals

    and , find ! #"%$&"(')"+* such that, .-!0/213454-!768% 9:#;?/A@3B-!>CD6 9&E!=!>?FG$H":' ":*(I (1)