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Numerical Methods for Differential Equations, Optimization, and Technological Problems pp 109–132Cite as

Balancing Discretization and Iteration Error in Finite Element A Posteriori Error Analysis

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Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 27))

Abstract

This article surveys recent developments in a combined a posteriori analysis for the discretization and iteration errors in the finite element approximation of elliptic PDE systems. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error control. Based on computable a posteriori error estimates the algebraic iteration can be adjusted to the discretization in a successive mesh adaptation process. The performance of the proposed method is demonstrated for several model situations including the simple Poisson equation, the Stokes equations in fluid mechanics and the KKT system of a linear-quadratic elliptic optimal control problem. Furthermore, extensions are discussed for certain classes of nonlinear problems including eigenvalue problems and nonlinear reaction-diffusion equations.

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Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, University of Heidelberg, INF 293/294, 69120, Heidelberg, Germany

    Rolf Rannacher & Jevgeni Vihharev

Authors
  1. Rolf Rannacher
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  2. Jevgeni Vihharev
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Corresponding author

Correspondence to Rolf Rannacher .

Editor information

Editors and Affiliations

  1. Mathematical Information Technology, University of Jyväskylä, Mattilanniemi 2, Jyväskylä, 40100, Finland

    Sergey Repin

  2. Mathematical Information Technology, University of Jyväskylä, Mattilanniemi 2, Jyväskylä, 40100, Finland

    Timo Tiihonen

  3. Mathematical Information Technology, University of Jyväskylä, Mattilanniemi 2, Jyväskylä, 40100, Finland

    Tero Tuovinen

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Rannacher, R., Vihharev, J. (2013). Balancing Discretization and Iteration Error in Finite Element A Posteriori Error Analysis. In: Repin, S., Tiihonen, T., Tuovinen, T. (eds) Numerical Methods for Differential Equations, Optimization, and Technological Problems. Computational Methods in Applied Sciences, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5288-7_6

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  • DOI: https://doi.org/10.1007/978-94-007-5288-7_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-5287-0

  • Online ISBN: 978-94-007-5288-7

  • eBook Packages: EngineeringEngineering (R0)

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