Open Access
Issue
ESAIM: M2AN
Volume 59, Number 5, September-October 2025
Page(s) 2837 - 2861
DOI https://doi.org/10.1051/m2an/2025077
Published online 24 October 2025
  1. N. Ahmed and G. Matthies, Higher order continuous Galerkin–Petrov time stepping schemes for transient convection-diffusion-reaction equations. ESAIM: Math. Model. Numer. Anal. 49 (2015) 1429–1450. [Google Scholar]
  2. N. Ahmed, G. Matthies, L. Tobiska and H. Xie, Discontinuous Galerkin time stepping with local projection stabilization for transient convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 200 (2011) 1747–1756. [Google Scholar]
  3. G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: Math. Model. Numer. Anal. 38 (2004) 261–289. [Google Scholar]
  4. S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979) 1085–1095. [Google Scholar]
  5. 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]
  6. A.K. Aziz and P. Monk, Continuous finite elements in space and time for the heat equation. Math. Comput. 52 (1989) 255–274. [Google Scholar]
  7. I. Babuška and T. Janik, The h-p version of the finite element method for parabolic equations: Part II. The h-p version in time. Numer. Methods Part. Differ. Equ. 6 (1990) 343–369. [Google Scholar]
  8. I. Babuška and M. Suri, The p and h-p versions of the finite element method, basic principles and properties. SIAM Rev. 36 (1994) 578–632. [CrossRef] [MathSciNet] [Google Scholar]
  9. M. Bause, F.A. Radu and U. Köcher, Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space. Numer. Math. 137 (2017) 773–818. [Google Scholar]
  10. J. Becker, A second order backward difference method with variable steps for a parabolic problem. BIT 38 (1998) 644–662. [Google Scholar]
  11. D. Braess, Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 3rd edition. Cambridge University Press, Cambridge (2007). [Google Scholar]
  12. H. Brunner and D. Schötzau, hp-discontinuous Galerkin time-stepping for Volterra integrodifferential equations. SIAM J. Numer. Anal. 44 (2006) 224–245. [CrossRef] [MathSciNet] [Google Scholar]
  13. G. Buffoni, A. Griffa and E. Zambianchi, Modelling of dispersion processes in a tide-forced flow. Hydrobiologia 393 (1999) 19–24. [Google Scholar]
  14. Z.M. Chen and Y. Liu, Efficient and parallel solution of high-order continuous time Galerkin for dissipative and wave propagation problems. SIAM J. Sci. Comput. 46 (2024) A2073–A2100. [Google Scholar]
  15. M. Delfour, W. Hager and F. Trochu, Discontinuous Galerkin methods for ordinary differential equations. Math. Comput. 36 (1981) 455–473. [Google Scholar]
  16. J. Douglas and T. Dupont, Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 7 (1970) 575–626. [CrossRef] [MathSciNet] [Google Scholar]
  17. Y.H. Du ang Z.M. Guo, The Stefan problem for the Fisher-KPP equation. J. Differ. Equ. 253 (2012) 996–1035. [Google Scholar]
  18. K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Modél. Math. Anal. Numér. 19 (1985) 611–643. [Google Scholar]
  19. A. Ern and J.-L. Guermond, Finite Elements III: First-Order and Time-Dependent PDEs. Springer, Cham (2021). [Google Scholar]
  20. A. Ern and M. Vohralík, A posteriori error estimation based on potential and flux reconstruction for the heat equation. SIAM J. Numer. Anal. 48 (2010) 198–223. [CrossRef] [MathSciNet] [Google Scholar]
  21. L. Evans, Partial Differential Equations. American Mathematical Society, Providence, RI (2010). [Google Scholar]
  22. F. Fakhar-Izadi and M. Dehghan, A spectral element method using the modal basis and its application in solving second-order nonlinear partial differential equations. Math. Methods Appl. Sci. 38 (2015) 478–504. [Google Scholar]
  23. E.H. Georgoulis, O. Lakkis and J.M. Virtanen, A posteriori error control for discontinuous Galerkin methods for parabolic problems. SIAM J. Numer. Anal. 49 (2011) 427–458. [Google Scholar]
  24. E.H. Georgoulis, O. Lakkis and T.P. Wihler, A posteriori error bounds for fully-discrete hp-discontinuous Galerkin time-stepping methods for parabolic problems. Numer. Math. 148 (2021) 363–368. [Google Scholar]
  25. B.L. Hulme, One-step piecewise polynomial Galerkin methods for initial value problems. Math. Comput. 26 (1972) 415–426. [Google Scholar]
  26. B.L. Hulme, Discrete Galerkin and related one-step methods for ordinary differential equations. Math. Comput. 26 (1972) 881–891. [Google Scholar]
  27. S. Hussain, F. Schieweck and S. Turek, Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation. J. Numer. Math. 19 (2011) 41–61. [Google Scholar]
  28. S. Hussain, F. Schieweck and S. Turek, A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations. Open Numer. Methods J. 4 (2012) 35–45. [Google Scholar]
  29. A. Ilyin, A. Laptev, M. Loss and S. Zelik, One-dimensional interpolation inequalities, Carlson–Landau inequalities, and magnetic Schrödinger operators. Int. Math. Res. Notes 2016 (2016) 1190–1222. [Google Scholar]
  30. J. Isenberg and C. Gutfinger, Heat transfer to a draining film. Int. J. Heat Transfer 16 (1972) 505–512. [Google Scholar]
  31. P. Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal. 15 (1978) 91–928. [Google Scholar]
  32. B. Janssen and T.P. Wihler, Computational comparison of continuous and discontinuous Galerkin time-stepping methods for nonlinear initial value problems. Lect. Notes Comput. Sci. Eng. 106 (2015) 103–114. [Google Scholar]
  33. L.S. Kimpton, J.P. Whiteley, S.L. Waters and J.M. Oliver, Approaches to myosin modelling in a two-phase flow model for cell motility. Phys. D 318/319 (2016) 34–49. [Google Scholar]
  34. O. Lakkis and C. Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comput. 75 (2006) 1627–1658. [Google Scholar]
  35. H. Li, C.Y. Du and Z.H. Zhao, Continuous space-time finite element method for a reaction-diffusion equation. Math. Numer. Sin. 39 (2017) 167–178. [Google Scholar]
  36. C. Makridakis and I. Babuška, On the stability of the discontinuous Galerkin method for the heat equation. SIAM J. Numer. Anal. 34 (1997) 389–401. [Google Scholar]
  37. H. Matsuzawa, A free boundary problem for the fisher-KPP equation with a given moving boundary. Commun. Pure Appl. Anal. 17 (2018) 1821–1852. [Google Scholar]
  38. S. Metcalfe and T.P. Wihler, Conditional a posteriori error bounds for high order discontinuous Galerkin time stepping approximations of semilinear heat models with blow-up. SIAM J. Sci. Comput. 44 (2022) A1337–A1357. [Google Scholar]
  39. A. Miranville and R. Quintanilla, A generalization of the Allen–Cahn equation. IMA J. Appl. Math. 80 (2015) 410–430. [Google Scholar]
  40. S. Nicaise and N. Soualem, A posteriori error estimates for a nonconforming finite element discretization of the heat equation. ESAIM: Math. Model. Numer. Anal. 39 (2005) 319–348. [Google Scholar]
  41. E.M. Rønquist and A.T. Patera, A Legendre spectral element method for the Stefan problem. Int. J. Numer. Methods Eng. 24 (1987) 2273–2299. [Google Scholar]
  42. N. Saito, Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations. IMA J. Numer. Anal. 41 (2021) 1267–1292. [Google Scholar]
  43. J.R. Salmon, J.A. Liggett and R.H. Gallager, Dispersion analysis in homogeneous lakes. Int. J. Numer. Methods Eng. 15 (1980) 1627–1642. [Google Scholar]
  44. F. Schieweck, A-stable discontinuous Galerkin–Petrov time discretization of higher order. J. Numer. Math. 18 (2010) 25–57. [Google Scholar]
  45. D. Schötzau and C. Schwab, An hp a priori error analysis of the DG time-stepping method for initial value problems. Calcolo 37 (2000) 207–232. [Google Scholar]
  46. 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]
  47. D. Schötzau and T.P. Wihler, A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations. Numer. Math. 115 (2010) 475–509. [Google Scholar]
  48. C. Schwab, p-and hp-Finite Element Methods. Oxford University Press, New York (1998). [Google Scholar]
  49. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Vol. 25 of Springer Series in Computational Mathematics, 2nd edition. Springer-Verlag, Berlin (2006). [Google Scholar]
  50. L.N. Wang, M.Z. Zhang, H.J. Tian and L.J. Yi, hp-version C1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations. IMA J. Numer. Anal. 45 (2025) 1455–1500. [Google Scholar]
  51. Y.C. Wei and L.J. Yi, An hp-version of the C0-continuous Galerkin time-stepping method for nonlinear second-order initial value problems. Adv. Comput. Math. 46 (2020) 56. [Google Scholar]
  52. T. Werder, K. Gerdes, D. Schötzau and C. Schwab, hp-discontinuous Galerkin time stepping for parabolic problems. Comput. Methods Appl. Mech. Eng. 190 (2001) 6685–6708. [Google Scholar]
  53. T.P. Wihler, An a priori error analysis of the hp-version of the continuous Galerkin FEM for nonlinear initial value problems. J. Sci. Comput. 25 (2005) 523–549. [Google Scholar]
  54. L.J. Yi, An L-error estimate for the h-p version continuous Petrov–Galerkin method for nonlinear initial value problems. East Asian J. Appl. Math. 5 (2015) 301–311. [Google Scholar]
  55. L.J. Yi and B.Q. Guo, An h-p version of the continuous Petrov–Galerkin finite element method for Volterra integrodifferential equations with smooth and nonsmooth kernels. SIAM J. Numer. Anal. 53 (2015) 2677–2704. [CrossRef] [MathSciNet] [Google Scholar]
  56. J. Zhu and J. X. Qiu, Local DG method using WENO type limiters for convection-diffusion problems. J. Comput. Phys. 230 (2011) 4353–4375. [Google Scholar]
  57. Z. Zlatev, R. Berkowicz and L.P. Prahm, Implementation of a variable stepsize variable formula in the time-integration part of a code for treatment of long-range transport of air pollutants. J. Comput. Phys. 55 (1984) 278–301. [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