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Error analysis of HDG approximations for elliptic variational inequality: obstacle problem

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Abstract

In this article, we study the HDG approximation for the obstacle problem, i.e., variational inequalities, with remarkable convergence properties. Using polynomials of degree k ≥ 0 for both the potential u and the flux q, we show that the approximations of the potential and flux converge in L2 with the optimal order of k + 1 . The approximate trace of the potential is proved to converge with optimal order k + 1 in L2. Finally, numerical results are presented to verify these theoretical results.

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References

  1. Friedman, A.: Variational Principles and Free-Boundary Problems. second edition. Robert E. Krieger Publishing Variational Inc., Malabar (1988)

  2. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications Pure and Applied Mathematics, vol. 88. Academic, New York (A Subsidiary of Harcourt Brace Jovanovich, Publishers) (1980)

  3. Rodrigues, J.-F.: Obstacle Problems in Mathematical Physics, volume 134 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam (1987). Notas de Matematica [Mathematical Notes], 114

    Google Scholar 

  4. Brezis, H., Stampacchia, G.: Sur la regularite de la solution d’inequations elliptiques. Bull. Soc. Math. France 96, 153–180 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lions, J.: Quelques Methodes De R6solution Des Probl+Mes Aux Limites Non Lin6aires. Dunod, Paris (1969)

    Google Scholar 

  6. Fichera, G.: Boundary value problems of elasticity with unilateral con- straints. Handbuch der Physik, Band VI a/2, Mechanics of Solids II Springer, 1972, pp. 391–424

  7. Frehse, J.: On the regularity of the solution of a second order variational inequality. Boll. Un. Mat. Ital.(4) 6, 312–315 (1972)

    MathSciNet  MATH  Google Scholar 

  8. Brezzi, F., Hager, W.W., Raviart, P.A.: Error estimates for the finite element solution of variational inequalities part i. primal theorey. Numer. Math. 28, 431–443 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hlavacek: Dual finite element analysis for unilateral boundary value problems. Apl. Mat. 22, 14–51 (1977)

    MathSciNet  MATH  Google Scholar 

  10. Brezzi, F., Hager, Raviart, P.A.: Error estimates for the finite element solution of variational inequalities part II. Mixed methods. Numer. Math. 31, 1–16 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kikuchi, N.: Convergence of a penalty method for variational inequalities. TICOM Report 79-16 the University of Texas at Austin (1979)

  12. Glowinski, R., Lions, J.-L., Tremolieres, R.: Numerical Analysis of Variational In- equalities, North-Holland, Amsterdam (1981)

  13. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    Book  MATH  Google Scholar 

  14. Hoppe, R.H.W., Kornhuber, R.: Adaptive multilevel methods for obstacle problems. SIAM J. Numer. Anal. 31, 301–323 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Blum, H., Suttmeier, F.-T.: Weighted error estimates for finite element solutions of variational inequalities. Computing 65, 119–134 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yan, N.: A posteriori error estimators of gradient recovery type for elliptic obstacle problems[J]. Adv. Comput. Math. 15(1-4), 333–361 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, F., Han, W., Cheng, X.L.: Discontinuous Galerkin methods for solving elliptic variational inequalities[J]. SIAM J. Numer. Anal. 48(2), 708–733 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hlavacek: Dual finite element analysis for elliptic problems with obstacles on the boundary

  19. Cockburn, B., Gopalakrishnan, J.: A characterization of hybridized mixed methods for second order elliptic Problems[J]. SIAM J. Numer. Anal. 42(1), 283–301 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cockburn, B., Dong, B., Guzman, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77, 1887–1916 (2008). MR2429868 (2009d:65166)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cockburn, B.: Unified hybridization of discontinuous galerkin, mixed and continuous galerkin methods for second order elliptic Problems[J]. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cockburn, B., Gopalakrishnan, J., Sayas, F.-J.: A projection-based error analysis of HDG methods. Math. Comput. 79, 1351–1367 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, B, Xie, X.: Analysis of a family of HDG methods for second order elliptic problems [J]. J. Comput. Appl. Math. 307, 37–51 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This subject is supported partially by NSFC No. 11672032.

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Authors and Affiliations

  1. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    M. Zhao, H. Wu & C. Xiong

Authors
  1. M. Zhao
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  2. H. Wu
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  3. C. Xiong
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Correspondence to C. Xiong.

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Zhao, M., Wu, H. & Xiong, C. Error analysis of HDG approximations for elliptic variational inequality: obstacle problem. Numer Algor 81, 445–463 (2019). https://doi.org/10.1007/s11075-018-0556-5

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  • DOI: https://doi.org/10.1007/s11075-018-0556-5

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