Open Access
Volume 56, Number 6, November-December 2022
Page(s) 2255 - 2296
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. [CrossRef] [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. [MathSciNet] [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. [CrossRef] [EDP Sciences] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [EDP Sciences] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [Google Scholar]
  10. T. Dupont, L2-estimates for Galerkin methods for second order hyperbolic equations. SIAM J. Numer. Anal. 10 (1973) 880–889. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [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. [CrossRef] [MathSciNet] [Google Scholar]
  33. M. Dobrowolski and R. Rannacher, Finite element methods for nonlinear elliptic systems of second order. Math. Nachr. 94 (1980) 155–172. [CrossRef] [MathSciNet] [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. [CrossRef] [MathSciNet] [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. [CrossRef] [Google Scholar]
  38. R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38 (1982) 437–445. [CrossRef] [MathSciNet] [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]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you