Journal of Scientific Computing

, Volume 70, Issue 1, pp 407–428 | Cite as

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

  • Yanmin Zhao
  • Pan Chen
  • Weiping Bu
  • Xiangtao Liu
  • Yifa Tang


Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order \(0<\alpha <1\). As to the conforming scheme, the spatial global superconvergence and temporal convergence order of \(O(h^2+\tau ^{2-\alpha })\) for both the original variable u in \(H^1\)-norm and the flux \(\vec {p}=\nabla u\) in \(L^2\)-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and \(\tau \) are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of \(\textit{EQ}_1^{\textit{rot}}\) nonconforming element, which manifests that convergence orders of \(O(h+\tau ^{2-\alpha })\) and \(O(h^2+\tau ^{2-\alpha })\) for the original variable u in broken \(H^1\)-norm and \(L^2\)-norm, respectively, and approximation for the flux \(\vec {p}\) converging with order \(O(h+\tau ^{2-\alpha })\) in \(L^2\)-norm. Numerical examples are provided to demonstrate the theoretical analysis.


Mixed finite element methods L1 method Time-fractional diffusion equation Unconditional stability  Superconvergence and convergence 



This research is supported by National Natural Science Foundation of China (Grant Nos. 11101381 and 11371357) and Outstanding Young Talents Training Plan by Xuchang University.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Yanmin Zhao
    • 1
  • Pan Chen
    • 2
  • Weiping Bu
    • 3
  • Xiangtao Liu
    • 4
  • Yifa Tang
    • 5
  1. 1.School of Mathematics and StatisticsXuchang UniversityXuchangChina
  2. 2.Division of Basic EducationHuainan Vocational & Technical CollegeHuainanChina
  3. 3.School of Mathematics and Computational ScienceXiangtan UniversityXiangtanChina
  4. 4.Cipher GroundNorth BrunswickUSA
  5. 5.LSEC, ICMSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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