Journal of Applied Mathematics

Optimal Convergence Rates of Moving Finite Element Methods for Space-Time Fractional Differential Equations

Xuemei Gao and Xu Han

Full-text: Open access

Abstract

This paper studies the moving finite element methods for the space-time fractional differential equations. An optimal convergence rate of the moving finite element method is proved for the space-time fractional differential equations.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 792912, 6 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808145

Digital Object Identifier
doi:10.1155/2013/792912

Mathematical Reviews number (MathSciNet)
MR3108940

Zentralblatt MATH identifier
06950874

Citation

Gao, Xuemei; Han, Xu. Optimal Convergence Rates of Moving Finite Element Methods for Space-Time Fractional Differential Equations. J. Appl. Math. 2013 (2013), Article ID 792912, 6 pages. doi:10.1155/2013/792912. https://projecteuclid.org/euclid.jam/1394808145


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References

  • M. Kirane, Y. Laskri, and N.-e. Tatar, “Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 488–501, 2005.
  • C. J. Budd, W. Huang, and R. D. Russell, “Moving mesh methods for problems with blow-up,” SIAM Journal on Scientific Computing, vol. 17, no. 2, pp. 305–327, 1996.
  • W. Huang, J. Ma, and R. D. Russell, “A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations,” Journal of Computational Physics, vol. 227, no. 13, pp. 6532–6552, 2008.
  • J. Ma, Y. Jiang, and K. Xiang, “Numerical simulation of blowup in nonlocal reaction-diffusion equations using a moving mesh method,” Journal of Computational and Applied Mathematics, vol. 230, no. 1, pp. 8–21, 2009.
  • J. Ma and Y. Jiang, “Moving mesh methods for blowup in reaction-diffusion equations with traveling heat source,” Journal of Computational Physics, vol. 228, no. 18, pp. 6977–6990, 2009.
  • J. Ma and Y. Jiang, “Moving collocation methods for time fractional differential equations and simulation of blowup,” Science China, vol. 54, no. 3, pp. 611–622, 2011.
  • W. Huang and R. D. Russell, Adaptive Moving Mesh Methods, vol. 174 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2011.
  • M. J. Baines, Moving Finite Elements, Monographs on Numerical Analysis, Oxford University Press, Oxford, UK, 1994.
  • T. Tang and J. Xu, Adaptive Computations: Theory and Algorithms, Science Press, Beijing, China, 2007.
  • Y. Jiang and J. Ma, “Moving finite element methods for time fractional partial differential equations,” Science China. Mathematics, vol. 56, no. 6, pp. 1287–1300, 2013.
  • J. Ma, J. Liu, and Z. Zhou, “Convergence analysis of moving finite element methods for space fractional differential equations,” Journal of Computational and Applied Mathematics, vol. 255, pp. 661–670, 2014.
  • R. E. Bank and R. F. Santos, “Analysis of some moving space-time finite element methods,” SIAM Journal on Numerical Analysis, vol. 30, no. 1, pp. 1–18, 1993.
  • T. Dupont, “Mesh modification for evolution equations,” Mathematics of Computation, vol. 39, no. 159, pp. 85–107, 1982.
  • T. F. Dupont and Y. Liu, “Symmetric error estimates for moving mesh Galerkin methods for advection-diffusion equations,” SIAM Journal on Numerical Analysis, vol. 40, no. 3, pp. 914–927, 2002.
  • V. J. Ervin and J. P. Roop, “Variational formulation for the stationary fractional advection dispersion equation,” Numerical Methods for Partial Differential Equations, vol. 22, no. 3, pp. 558–576, 2006.
  • V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, vol. 25 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, Germany, 2nd edition, 2006.
  • J. A. Mackenzie and W. R. Mekwi, “An analysis of stability and convergence of a finite-difference discretization of a model parabolic PDE in 1D using a moving mesh,” IMA Journal of Numerical Analysis, vol. 27, no. 3, pp. 507–528, 2007.
  • J. Ma, Y. Jiang, and K. Xiang, “On a moving mesh method for solving partial integro-differential equations,” Journal of Computational Mathematics, vol. 27, no. 6, pp. 713–728, 2009.
  • C. de Boor, “Good approximation by splines with variable knots,” in Spline Functions and Approximation Theory, A. Meir and A. Sharma, Eds., pp. 57–73, Birkhäuser, Basel, Switzerland, 1973.