Open Access
Volume 57, Number 3, May-June 2023
Page(s) 1297 - 1322
Published online 12 May 2023
  1. G. Akrivis, C. Makridakis and R.H. Nochetto, Galerkin and Runge-Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence. Numer. Math. 118 (2011) 429–456. [Google Scholar]
  2. R. Andreev and J. Schweitzer, Conditional space-time stability of collocation Runge-Kutta for parabolic evolution equations. Electron. Trans. Numer. Anal. 41 (2014) 62–80. [MathSciNet] [Google Scholar]
  3. J.W. Barrett and J.F. Blowey, Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. Numer. Math. 77 (1997) 1–34. [Google Scholar]
  4. J.W. Barrett and J.F. Blowey, Finite element approximation of the Cahn-Hilliard equation withconcentration dependent mobility. Math. Comp. 68 (1999) 487–517. [Google Scholar]
  5. J.W. Barrett, J.F. Blowey and H. Garcke, Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Math. 80 (1998) 525–556. [Google Scholar]
  6. J.W. Barrett, J.F. Blowey and H. Garcke, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286–318. [Google Scholar]
  7. J.W. Barrett, J.F. Blowey and H. Garcke, On fully practical finite element approximations of degenerate Cahn-Hilliard systems. M2AN Math. Model. Numer. Anal. 35 (2001) 713–748. [Google Scholar]
  8. F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20 (1999) 175–212. [MathSciNet] [Google Scholar]
  9. L.M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming. Comput. Math. Math. Phys. 7 (1967) 200–217. [CrossRef] [Google Scholar]
  10. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Vol. 15 of Texts in Applied Mathematics, 3rd edition. Springer, New York, NY (2008). [Google Scholar]
  11. A. Brunk, B. Dünweg, H. Egger, O. Habrich, M. Lukáčová-Medvid’ová and D. Spiller, Analysis of a viscoelastic phase separation model. J. Phys. Condens. Matter. 11 (2021) 33. [Google Scholar]
  12. J.W. Cahn, On spinodal decomposition. Acta Metall. 9 (1961) 795–801. [CrossRef] [Google Scholar]
  13. J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system I. interfacial free energy. J. Chem. Phys. 28 (1958) 258–267. [CrossRef] [Google Scholar]
  14. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Vol. 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). [Google Scholar]
  15. M.I.M. Copetti and C.M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math. 63 (1992) 39–65. [Google Scholar]
  16. A.E. Diegel, C. Wang and S.M. Wise, Stability and convergence of a second-order mixed finite element method for the Cahn-Hilliard equation. IMA J. Numer. Anal. 36 (2016) 1867–1897. [CrossRef] [MathSciNet] [Google Scholar]
  17. Q. Du and R.A. Nicolaides, Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28 (1991) 1310–1322. [Google Scholar]
  18. C.M. Elliott and D.A. French, A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 26 (1989) 884–903. [Google Scholar]
  19. C.M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (1996) 404–423. [Google Scholar]
  20. C.M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation. Math. Comp. 58 (1992) 603–630 S33–S36. [Google Scholar]
  21. C.M. Elliott, D.A. French and F.A. Milner, A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54 (1989) 575–590. [Google Scholar]
  22. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). [CrossRef] [Google Scholar]
  23. E. Feireisl, M. Lukáčová-Medvid’ová, S. Nečasová, A. Novotná and B. She, Asymptotic preserving error estimates for numerical solutions of compressible Navier-Stokes equations in the low mach number regime. SIAM Multiscale Model. Simul. 16 (2018) 150–183. [Google Scholar]
  24. E. Feireisl, M. Lukáčová-Medvid’ová, H. Mizerová and B. She, Numerical Analysis of Compressible Fluid Flows, Vol. 20 of MS&A. Modeling, Simulation and Applications. Springer, Cham (2021). [Google Scholar]
  25. E. Feireisl, M. Lukáčová-Medvid’ová and B. She, Improved error estimates for the finite volume and the MAC schemes for the compressible Navier-Stokes system. Numer. Math. (2023). DOI: 10.48550/arXiv.2205.04076. [Google Scholar]
  26. X. Feng, Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44 (2006) 1049–1072. [Google Scholar]
  27. X. Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99 (2004) 47–84. [Google Scholar]
  28. X. Feng and A. Prohl, Numerical analysis of the Cahn-Hilliard equation and approximation of the Hele-Shaw problem. Interfaces Free Bound. 7 (2005) 1–28. [CrossRef] [MathSciNet] [Google Scholar]
  29. T. Gallouët, R. Herbin, D. Maltese and A. Novotny, Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations. IMA J. Numer. Anal. 36 (2016) 543–592. [CrossRef] [MathSciNet] [Google Scholar]
  30. A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations. Springer (2016). [Google Scholar]
  31. D. Kay, V. Styles and E. Süli, Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection. SIAM J. Numer. Anal. 47 (2009) 2660–2685. [Google Scholar]
  32. D. Li and Z. Qiao, On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations. J. Sci. Comput. 70 (2017) 301–341. [Google Scholar]
  33. C. Liu, F. Frank and B.M. Rivière, Numerical error analysis for nonsymmetric interior penalty discontinuous Galerkin method of Cahn-Hilliard equation. Numer. Methods Part. Differ. Equ. 35 (2019) 1509–1537. [Google Scholar]
  34. M. Lukáčová-Medvid’ová, P.J. Strasser, B. Dünweg and N. Tretyakov, Energy-stable numerical schemes for multiscale simulations of polymer-solvent mixtures, in Mathematical Analysis of Continuum Mechanics and Industrial Applications II, edited by P. van Meurs, M. Kimura, H. Notsu. Springer Singapore (2018) 153–165. [Google Scholar]
  35. M. Lukáčová-Medvid’ová, B. She and Y. Yuan, Error estimates of the Godunov method for the multidimensional compressible Euler system. J. Sci. Comput. 91 (2022) 27. [Google Scholar]
  36. N. Meyers and J. Serrin, H = W. Proc. Nat. Acad. Sci. USA 51 (1964) 1055–1056. [Google Scholar]
  37. P.J. Strasser, G. Tierra, B. Dünweg and M. Lukáčová-Medvid’ová, Energy-stable linear schemes for polymer-solvent phase field models. Comput. Math. Appl. 77 (2019) 125–143. [CrossRef] [MathSciNet] [Google Scholar]
  38. G. Tierra and F. Guillén-González, Numerical methods for solving the Cahn-Hilliard equation and its applicability to related energy-based models. Arch. Comput. Methods Eng. 22 (2015) 269–289. [CrossRef] [MathSciNet] [Google Scholar]
  39. J. Wloka, Partial Differential Equations. Translated from the German by C.B. Thomas and M.J. Thomas. Cambridge University Press, Cambridge (1987). [Google Scholar]
  40. Y. Xia, Y. Xu and C.-W. Shu, Local discontinuous Galerkin methods for the Cahn-Hilliard type equations. J. Comput. Phys. 227 (2007) 472–491. [CrossRef] [MathSciNet] [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