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Discontinuous Galerkin methods for solving a quasistatic contact problem

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Abstract

We consider the numerical solution of a nonlinear evolutionary variational inequality, arising in the study of quasistatic contact problems. We study spatially semi-discrete and fully discrete schemes for the problem with several discontinuous Galerkin discretizations in space and finite difference discretization in time. Under appropriate regularity assumptions on the solution, a unified error analysis is established for the schemes, reaching the optimal convergence order for linear elements. Numerical results are presented on a two dimensional test problem to illustrate numerical convergence orders.

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References

  1. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arnold, D.N., Brezzi, F., Marini, L.D.: A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate. J. Sci. Comput. 22, 25–45 (2005)

    Article  MathSciNet  Google Scholar 

  3. Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, Berlin (2009)

    Google Scholar 

  4. Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., Savini, M.: A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In: Decuypere, R., Dibelius, G. (eds.) Proceedings of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics, pp. 99–108. Technologisch Instituut, Antwerpen, Belgium (1997)

  5. Brenner, S.C.: Korn’s inequalities for piecewise \(H^1\) vector fields. Math. Comput. 73, 1067–1087 (2004)

    Article  MATH  Google Scholar 

  6. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  7. Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous finite elements for diffusion problems. In: Atti Convegno in onore di F. Brioschi (Milan, 1997) Istituto Lombardo. Accademia di Scienze e Lettere, Milan, Italy, pp. 197–217 (1999)

  8. Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Equ. 16, 365–378 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)

    MATH  Google Scholar 

  10. Cockburn, B., Karniadakis, G. E., Shu, C.-W. (eds): Discontinuous Galerkin Methods. Theory, Computation and Applications, Lecture Notes in Computer Science engineering, vol. 11. Springer, New York (2000)

  11. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Djoko, J.K., Ebobisse, F., McBride, A.T., Reddy, B.D.: A discontinuous Galerkin formulation for classical and gradient plasticity—part 1: formulation and analysis. Comput. Methods Appl. Mech. Eng. 196, 3881–3897 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Djoko, J.K., Ebobisse, F., McBride, A.T., Reddy, B.D.: A discontinuous Galerkin formulation for classical and gradient plasticity—part 2: algorithms and numerical analysis. Comput. Methods Appl. Mech. Eng. 197, 1–21 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Douglas, Jr, J., Dupont, T.: Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods, Lecture Notes in Physics, vol. 58. Springer, Berlin (1976)

  15. Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. American Mathematical Society and International Press, USA (2002)

    MATH  Google Scholar 

  16. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)

    Book  MATH  Google Scholar 

  17. Wang, F., Han, W., Cheng, X.: Discontinuous Galerkin methods for solving elliptic variational inequalities. SIAM J. Numer. Anal. 48, 708–733 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wang, F., Han, W., Cheng, X.: Discontinuous Galerkin methods for solving Signorini problem. IMA J. Numer. Anal. 31, 1754–1772 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wang, F., Han, W., Eichholz, J.: A posteriori error estimates of discontinuous Galerkin methods for obstacle problems (submitted)

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Acknowledgments

We thank the two anonymous referees for their valuable comments that lead to an improvement of the paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Fei Wang

  2. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Fei Wang & Weimin Han

  3. Program in Applied Mathematical and Computational Sciences, University of Iowa, Iowa City, IA, 52242, USA

    Weimin Han

  4. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Xiaoliang Cheng

Authors
  1. Fei Wang
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  2. Weimin Han
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  3. Xiaoliang Cheng
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Corresponding author

Correspondence to Fei Wang.

Additional information

F. Wang was supported by the Chinese National Science Foundation (Grant No. 11101168). W. Han was supported by grants from the Simons Foundation. X. Cheng was supported by the Key Project of the Major Research Plan of NSFC (Grant No. 91130004).

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Wang, F., Han, W. & Cheng, X. Discontinuous Galerkin methods for solving a quasistatic contact problem. Numer. Math. 126, 771–800 (2014). https://doi.org/10.1007/s00211-013-0574-0

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  • DOI: https://doi.org/10.1007/s00211-013-0574-0

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