Moving finite element methods for time fractional partial differential equations

Science China Mathematics

Volumn 56, Issue 6, 2013, Pages 1287-1300

  Jiang, YingJun ^a   Ma, JingTang ^b  

^a CHANGSHA UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

^b SOUTHWESTERN UNIVERSITY OF FINANCE AND ECONOMICS   (China)

Author keywords

blow up solutions; fractional partial differential equations; moving finite element methods

Indexed keywords

EID: 84878803579     PISSN: 16747283     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11425-013-4584-2     Document Type: Article

References (22)

1. Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods. New York: Springer-Verlag, 2002.
2. Chen C, Liu F, Turner I, et al. A fourier method for the fractional diffusion equation describing sub-diffusion. J Comput Phys, 2007, 227: 886-897.
3. Diethelm K, Ford N J, Freed A D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynam, 2002, 29: 3-22.
4. Diethelm K, Ford N J, Freed A D. Detailed error analysis for a fractional Adams method. Numer Algorithm, 2004, 36: 31-52.
5. Gorenflo R, Mainardi F. Some recent advances in theory and simulation of fractional diffusion processes. J Comput Appl Math, 2009, 229: 400-415.
6. Huang W, Ren Y, Russell R D. Moving mesh partial differential equations (MMPDEs) based upon the equidistribution principle. SIAM J Numer Anal, 1994, 31: 709-730.
7. Huang W, Russell R D. Adaptive Moving Mesh Methods. New York: Springer-Verlag, 2011.
8. Huang W Z, Sun W W. Variational mesh adaptation II: error estimates and monitor functions. J Comput Phys, 2003, 184: 619-648.
9. Jiang Y J, Ma J T. High-order finite element methods for time-fractional partial differential equations. J Comput Appl Math, 2011, 235: 3285-3290.
10. Kirane M, Laskri Y, Tatar N E. Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives. J Math Anal Appl, 2005, 312: 488-501.
11. Li C, Deng W. Remarks on fractional derivatives. Appl Math Comput, 2007 187: 777-784.
12. Li X J, Xu C J. A space-time spectral method for the time fractional diffusion equation. SIAM J Numer Anal, 2009 47: 2108-2131.
13. Lin Y M, Xu C J. Finite difference/spectral spproximations for the time-fractional diffusion equation. J Comput Phys, 2007, 225: 1533-1552.
14. Liu Y X, Shu C W. Error analysis of the semi-discrete local discontinuous Galerkin method for semiconductor device simulation models. Sci China Math, 2010, 12: 3255-3278.
15. Lubich C. Discretized fractional calculus. SIAM J Math Anal, 1986, 17: 704-719.
16. Ma J T, Jiang Y J. Moving collocation methods for time fractional differential equations and simulation of blowup. Sci China Math, 2011, 54: 611-622.
17. McLean W, Mustapha K. Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation. Numer Algorithm, 2009, 5: 69-88.
18. Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep, 2000, 339: 1-77.
19. Mustapha K, McLean M. Piecewise-linear, discontinous Galerkin method for a fractional diffusion equation. Numer Algorithm, 2011, 56: 159-184.
20. Yang J M, Chen Y P. A priori error analysis of a discontinuous Galerkin approximation for a kind of compressible miscible displacement problems. Sci China Math, 2010, 10: 2679-2696.
21. Zhuang P, Liu F, Anh V, et al. New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J Numer Anal, 2008, 46: 1079-1095.
22. Zhuang P, Liu F, Anh V, et al. Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process. IMA J Numer Math, 2009, 74: 645-667.