Numerische Mathematik

, Volume 141, Issue 3, pp 715–742 | Cite as

Convergence of \(\text{ dG(1) }\) in elastoplastic evolution

  • Carsten Carstensen
  • Dongjie LiuEmail author
  • Jochen Alberty


The discontinuous Galerkin (dG) methodology provides a hierarchy of time discretization schemes for evolutionary problems such as elastoplasticity with the Prandtl-Reuß  flow rule. A dG time discretization has been proposed for a variational inequality in the context of rate-independent inelastic material behaviour in Alberty and Carstensen in (CMME 191:4949–4968, 2002) with the help of duality in convex analysis to justify certain jump terms. This paper establishes the first a priori error analysis for the dG(1) scheme with discontinuous piecewise linear polynomials in the temporal and lowest-order finite elements for the spatial discretization. Compared to a generalized mid-point rule, the dG(1) formulation distributes the action of the material law in the form of the variational inequality in time and so it introduces an error in the material law. This may result in a suboptimal convergence rate for the dG(1) scheme and this paper shows that the stress error in the \(L^\infty (L^2)\) norm is merely \(O(h+k^{3/2})\) based on a seemingly sharp error analysis. The numerical investigation for a benchmark problem with known analytic solution provides empirical evidence of a higher convergence rate of the dG(1) scheme compared to dG(0).

Mathematics Subject Classification

65N30 65R20 73C50 



The work has been written, while the authors enjoyed the hospitality of the Hausdorff Research Institute of Mathematics in Bonn, Germany, during the Hausdorff Trimester Program Multiscale Problems: Algorithms, Numerical Analysis and Computation. The research of the first author (CC) has been supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 ‘Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis’ under the project ‘Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics’ (CA 151/22-1). The corresponding author’s research was supported by National Natural Science Foundation of China (Nos. 11571226, 11671313). The authors thank Simone Hock for a collaboration at the early stage of this research and the anonymous referees for valuable remarks that improved the presentation.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Carsten Carstensen
    • 1
  • Dongjie Liu
    • 2
    Email author
  • Jochen Alberty
    • 3
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Department of Mathematics, College of SciencesShanghai UniversityShanghaiPeople’s Republic of China
  3. 3.FrankfurtGermany

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