Open Access
Issue
ESAIM: M2AN
Volume 56, Number 6, November-December 2022
Page(s) 2255 - 2296
DOI https://doi.org/10.1051/m2an/2022066
Published online 08 December 2022
  1. T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations. Springer (1975) 25–70. [Google Scholar]
  2. T.J.R. Hughes, T. Kato and J.E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Ration. Mech. Anal. 63 (1976) 273–294. [Google Scholar]
  3. C.M. Dafermos and W.J. Hrusa, Energy methods for quasilinear hyperbolic initial-boundary value problems: applications to elastodynamics. Arch. Ration. Mech. Anal. 87 (1985) 267–292. [CrossRef] [Google Scholar]
  4. C.P. Chen and W. von Wahl, Das Rand-Anfangswertproblem für quasilineare Wellengleichungen in Sobolevräumen niedriger Ordnung. J. Reine Angew. Math. 337 (1982) 77–112. [Google Scholar]
  5. G.A. Baker, V.A. Dougalis and S.M. Serbin, High order accurate two-step approximations for hyperbolic equations. RAIRO Anal. Numér. 13 (1979) 201–226. [Google Scholar]
  6. L.A. Bales, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients. Math. Comp. 43 (1984) 383–414. [Google Scholar]
  7. G.A. Baker and J.H. Bramble, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations. RAIRO Anal. Numér. 13 (1979) 75–100. [Google Scholar]
  8. L.A. Bales and V.A. Dougalis, Cosine methods for nonlinear second-order hyperbolic equations. Math. Comp. 52299–319 (1989) S15–S33. [Google Scholar]
  9. L.A. Bales, Higher order single step fully discrete approximations for second order hyperbolic equations with time dependent coefficients. SIAM J. Numer. Anal. 23 (1986) 27–43. [Google Scholar]
  10. T. Dupont, L2-estimates for Galerkin methods for second order hyperbolic equations. SIAM J. Numer. Anal. 10 (1973) 880–889. [Google Scholar]
  11. J.E. Dendy Jr., Galerkin’s method for some highly nonlinear problems. SIAM J. Numer. Anal. 14 (1977) 327–347. [Google Scholar]
  12. C.G. Makridakis, Finite element approximations of nonlinear elastic waves. Math. Comp. 61 (1993) 569–594. [Google Scholar]
  13. C. Ortner and E. Süli, Discontinuous Galerkin finite element approximation of nonlinear second-order elliptic and hyperbolic systems. SIAM J. Numer. Anal. 45 (2007) 1370–1397. [Google Scholar]
  14. M. Hochbruck and B. Maier, Error analysis for space discretizations of quasilinear wave-type equations. IMA J. Numer. Anal. 42 (2021) 1963–1990. [Google Scholar]
  15. S.S. Antman, Nonlinear Problems of Elasticity 2nd edition. Vol. 107 of Applied Mathematical Sciences. Springer, , New York (2005). [Google Scholar]
  16. M.E. Gurtin, An Introduction to Continuum Mechanics. Vol. 158 of Mathematics in Science and Engineering. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London (1981). [Google Scholar]
  17. W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation. Tech. report, Los Alamos Scientific Lab., N. Mex. (USA) (1973). [Google Scholar]
  18. P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations: Proceedings of a Symposium Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, 1974. Academic Press (2014). [Google Scholar]
  19. I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973) 863–875. [Google Scholar]
  20. G.A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31 (1977) 45–59. [CrossRef] [MathSciNet] [Google Scholar]
  21. M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152–161. [Google Scholar]
  22. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. [Google Scholar]
  23. B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM (2008). [CrossRef] [Google Scholar]
  24. P.F. Antonietti, I. Mazzieri, M. Muhr, V. Nikolić and B. Wohlmuth, A high-order discontinuous Galerkin method for nonlinear sound waves. J. Comput. Phys. 415 (2020) 109484. [CrossRef] [MathSciNet] [Google Scholar]
  25. M. Muhr, B. Wohlmuth and V. Nikolić, A discontinuous Galerkin coupling for nonlinear elasto-acoustics, IMA J. Numer. Anal. . Preprint arXiv:2102.04311 (2021). [Google Scholar]
  26. P.F. Antonietti, I. Mazzieri, N. Dal Santo and A. Quarteroni, A high-order discontinuous Galerkin approximation to ordinary differential equations with applications to elastodynamics. IMA J. Numer. Anal. 38 (2018) 1709–1734. [CrossRef] [MathSciNet] [Google Scholar]
  27. C. Ortner and E. Süli, Discontinuous Galerkin finite element approximation of nonlinear second-order elliptic and hyperbolic systems. Tech. report, Oxford University Computing Laboratory, London (2006). [Google Scholar]
  28. K. Zhang, On the coercivity of elliptic systems in two-dimensional spaces. Bull. Aust. Math. Soc. 54 (1996) 423–430. [CrossRef] [Google Scholar]
  29. V. Šverák, Rank-one convexity does not imply quasiconvexity. Proc. R. Soc. Edinburgh Sect. A 120 (1992) 185–189. [Google Scholar]
  30. B. Dacorogna, P. Marcellini,A counterexample in the vectorial calculus of variations, in Material Instabilities in Continuum Mechanics (Edinburgh, 1985–1986). Oxford Sci. Publ., Oxford Univ. Press, New York (1988) 77–83. [Google Scholar]
  31. C.B. Morrey Jr., Multiple Integrals in the Calculus of Variations. Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York (1966). [Google Scholar]
  32. C. Bernardi, Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26 (1989) 1212–1240. [Google Scholar]
  33. M. Dobrowolski and R. Rannacher, Finite element methods for nonlinear elliptic systems of second order. Math. Nachr. 94 (1980) 155–172. [Google Scholar]
  34. R. Rannacher, On finite element approximation of general boundary value problems in nonlinear elasticity. Calcolo 17 (1980) 175–193. [CrossRef] [MathSciNet] [Google Scholar]
  35. D. Schötzau and C. Schwab, Time discretization of parabolic problems by the hp-version of the discontinuous Galerkin finite element method. SIAM J. Numer. Anal. 38 (2000) 837–875. [Google Scholar]
  36. A. Shao, A high-order discontinuous galerkin in time discretization for second-order hyperbolic equations. Preprint arXiv:2111.14642 (2021). [Google Scholar]
  37. Y.-Z. Chen and L.-C. Wu, Second Order Elliptic Equations and Elliptic Systems. Vol. 174 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1998). Translated from the 1991 Chinese original by Bei Hu. [Google Scholar]
  38. R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38 (1982) 437–445. [Google Scholar]
  39. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edition. Vol. 15 of Texts in Applied Mathematics. Springer, New York (2008). [Google Scholar]
  40. C.G. Makridakis, Galerkin/finite element methods for the equations of elastodynamics. Ph.D. thesis, Univ. of Cretex, Greek (1989). [Google Scholar]
  41. C.G. Makridakis, Finite element approximations of nonlinear elastic waves. Tech. report, Dept. of Mathematics, Univ. of Crete (1992). [Google Scholar]

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