Duality Techniques for Error Estimation and Mesh Adaptation in Finite Element Methods

  • Rolf Rannacher
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 416)


We present a general method for error control and mesh adaptivity in Galerkin finite element discretization of variational problems governed by differential equations. Our approach is based on the variational framework of projection methods and uses concepts from optimal control and sensitivity analysis. By employing global duality arguments and Galerkin orthogonality, we derive a posteriori error estimates in approximating quantities of physical interest such as, for example, point- or mean-values of boundary stresses. In these estimates the cellwise residuals of the computed solution are multiplied by weights which are obtained from the approximate solution of a dual problem. In this way, wo obtain the basis of a feed-back process by which the mesh is successively adapted, i.e. locally refined or coarsened, according to the particular goal of the computation. This method is systematically developed and analized at first for linear elliptic, parabolic as well as hyperbolic model problems. Then, it is extended to nonlinear problems with an application to the Hencky model of static elasto-plasticity. In the last two sections, we present applications in acoustics governed by the linear wave equation and for a nonlinear boundary control problem occuring in super-conductivity.


Optimal Control Problem Dual Problem Posteriori Error Dual Solution Posteriori Error Estimate 
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Copyright information

© CISM, Udine 2005

Authors and Affiliations

  • Rolf Rannacher
    • 1
  1. 1.Institut für Angewandte MathematikUniversität Heidelberg INF 293HeidelbergGermany

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