Discontinuous Galerkin Methods for Hemivariational Inequalities in Contact Mechanics (original) (raw)

References

  1. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)
    MathSciNet MATH Google Scholar
  2. 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)
    MathSciNet MATH Google Scholar
  3. Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, New York (2009)
    MATH Google Scholar
  4. Barboteu, M., Bartosz, K., Han, W., Janiczko, T.: Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact. SIAM J. Numer. Anal. 53, 527–550 (2015)
    MathSciNet MATH Google Scholar
  5. Barboteu, M., Bartosz, K., Kalita, P.: An analytical and numerical approach to a bilateral contact problem with nonmonotone friction. Int. J. Appl. Math. Comput. Sci. 23, 263–276 (2013)
    MathSciNet MATH Google Scholar
  6. Barboteu, M., Bartosz, K., Kalita, P., Ramadan, A.: Analysis of a contact problem with normal compliance, finite penetration and nonmonotone slip dependent friction. Commun. Contemp. Math. 16, 1350016 (2014)
    MathSciNet MATH Google Scholar
  7. 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. Technologisch Instituut, Antwerpen, Belgium, pp. 99–108 (1997)
  8. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer-Verlag, New York (2008)
    MATH Google Scholar
  9. Brenner, S.C., Sung, L., Zhang, H., Zhang, Y.: A quadratic \(C^0\) interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates. SIAM J. Numer. Anal. 50, 3329–3350 (2012)
    MathSciNet MATH Google Scholar
  10. 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,: Istituto Lombardo. Accademia di Scienze e Lettere, Milan, Italy 1999, 197–217 (1997)
  11. Buffa, A., Ortner, C.: Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal. 29, 827–855 (2009)
    MathSciNet MATH Google Scholar
  12. Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications. Springer, New York (2007)
    MATH Google Scholar
  13. Cascavita, K.L., Chouly, F., Ern, A.: Hybrid high-order discretizations combined with Nitsche’s method for Dirichlet and Signorini boundary conditions. IMA J. Numer. Anal. 40, 2189–2226 (2020)
    MathSciNet MATH Google Scholar
  14. Chouly, F., Ern, A., Pignet, N.: A hybrid high-order discretization combined with Nitsche’s method for contact and Tresca friction in small strain elasticity. SIAM J. Sci. Comput. 42, 2300–2324 (2020)
    MathSciNet MATH Google Scholar
  15. Chouly, F., Fabre, M., Hild, P., Pousin, J., Renard, Y.: Residual-based a posteriori error estimation for contact problems approximated by Nitsche’s method. IMA J. Numer. Anal. 38, 921–954 (2018)
    MathSciNet MATH Google Scholar
  16. Chouly, F., Hild, P.: A Nitsche-based method for unilateral contact problems: numerical analysis. SIAM J. Numer. Anal. 51, 1295–1307 (2013)
    MathSciNet MATH Google Scholar
  17. Chouly, F., Mlika, R., Renard, Y.: An unbiased Nitsche’s approximation of the frictional contact between two elastic structures. Numer. Math. 139, 593–631 (2018)
    MathSciNet MATH Google Scholar
  18. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
    MATH Google Scholar
  19. Clarke, F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)
    MathSciNet MATH Google Scholar
  20. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
    MATH Google Scholar
  21. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1997)
    MathSciNet MATH Google Scholar
  22. Cockburn, B., Karniadakis, G. E., Shu, C.-W.: Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering 11. Springer-Verlag, New York, 1119–1148 (2000)
  23. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston (2003)
    MATH Google Scholar
  24. 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)
    MATH Google Scholar
  25. Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)
    MATH Google Scholar
  26. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)
    MATH Google Scholar
  27. Guan, Q., Gunzburger, M., Zhao, W.: Weak-Galerkin finite element methods for a second-order elliptic variational inequality. Comput. Methods Appl. Mech. Eng. 337, 677–688 (2018)
    MathSciNet MATH Google Scholar
  28. Gudi, T., Porwal, K.: A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems. Math. Comp. 83, 579–602 (2014)
    MathSciNet MATH Google Scholar
  29. Han, W., Migórski, S., Sofonea, M.: A class of variational–hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891–3912 (2014)
    MathSciNet MATH Google Scholar
  30. Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30. Americal Mathematical Society, Providence (2002)
    MATH Google Scholar
  31. Han, W., Sofonea, M.: Numerical analysis of hemivariational inequalities in contact mechanics. Acta Numer. 28, 175–286 (2019)
    MathSciNet MATH Google Scholar
  32. Han, W., Sofonea, M., Barboteu, M.: Numerical analysis of elliptic hemivariational inequalities. SIAM J. Numer. Anal. 55, 640–663 (2017)
    MathSciNet MATH Google Scholar
  33. Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities, Methods and Applications. Kluwer Academic Publishers, Boston (1999)
    MATH Google Scholar
  34. Hong, Q., Wang, F., Wu, S., Xu, J.: A unified study of continuous and discontinuous Galerkin methods. Sci. China Math. 62, 1–32 (2019)
    MathSciNet MATH Google Scholar
  35. Jing, F., Han, W., Yan, W., Wang, F.: Discontinuous Galerkin finite element methods for stationary Navier-Stokes problem with a nonlinear slip boundary condition of friction type. J. Sci. Comput. 76, 888–912 (2018)
    MathSciNet MATH Google Scholar
  36. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)
    MATH Google Scholar
  37. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics 26, Springer, New York (2013)
  38. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc, New York (1995)
    MATH Google Scholar
  39. Panagiotopoulos, P.D.: Nonconvex problems of semipermeable media and related topics. Z. Angew. Math. Mech. 65, 29–36 (1985)
    MathSciNet MATH Google Scholar
  40. Panagiotopoulos, P.D.: Hemivariational Inequalities. Applications in Mechanics and Engineering. Springer, Berlin (1993)
    MATH Google Scholar
  41. Rivière, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I. Comput. Geosci. 3, 337–360 (1999)
    MathSciNet MATH Google Scholar
  42. Sofonea, M., Migórski, S.: Variational–Hemivariational Inequalities with Applications. Chapman & Hall/CRC Press, Boca Raton (2018)
    MATH Google Scholar
  43. Wang, F., Han, W.: Discontinuous Galerkin methods for solving a hyperbolic variational inequality. Numer. Methods PDEs 35, 894–915 (2019)
    MATH Google Scholar
  44. Wang, F., Han, W., Cheng, X.: Discontinuous Galerkin methods for solving elliptic variational inequalities. SIAM J. Numer. Anal. 48, 708–733 (2010)
    MathSciNet MATH Google Scholar
  45. Wang, F., Han, W., Cheng, X.: Discontinuous Galerkin methods for solving Signorini problem. IMA J. Numer. Anal. 31, 1754–1772 (2011)
    MathSciNet MATH Google Scholar
  46. Wang, F., Han, W., Cheng, X.: Discontinuous Galerkin methods for solving a quasistatic contact problem. Numer. Math. 126, 771–800 (2014)
    MathSciNet MATH Google Scholar
  47. Wang, F., Ling, M., Han, W., Jing, F.: Adaptive discontinuous Galerkin methods for solving an incompressible Stokes flow problem with slip boundary condition of frictional type. J. Comput. Appl. Math. 371, 112270 (2020)
    MathSciNet MATH Google Scholar
  48. Wang, F., Qi, H.: A discontinuous Galerkin method for an elliptic hemivariational inequality for semipermeable media. Appl. Math. Lett. 109, 106572 (2020)
    MathSciNet MATH Google Scholar
  49. Wang, F., Shah, S., Xiao, W.: A priori error estimates of discontinuous Galerkin methods for a quasi-variational inequality. BIT Numer. Anal. (2021). https://doi.org/10.1007/s10543-021-00848-1
    Article MATH Google Scholar
  50. Xiao, W., Wang, F., Han, W.: Discontinuous Galerkin methods for solving a frictional contact problem with normal compliance. Numer. Funct. Anal. Optim. 39, 1248–1264 (2018)
  51. Zeng, Y., Chen, J., Wang, F.: Error estimates of the weakly over-penalized symmetric interior penalty method for two variational inequalities. Comput. Math. Appl. 69, 760–770 (2015)
    MathSciNet MATH Google Scholar

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