J Sci Comput (2017) 72:917–935
Superconvergence of Finite Element Approximations
for the Fractional Diffusion-Wave Equation
· Xiaonian Long
· Shipeng Mao
Received: 5 August 2016 / Revised: 5 December 2016 / Accepted: 9 February 2017 /
Published online: 21 February 2017
© Springer Science+Business Media New York 2017
Abstract In this paper, the error estimates of fully discrete ﬁnite element approximation for
the time fractional diffusion-wave equation are discussed. Based on the standard Galerkin
ﬁnite element method approach for the spatial discretization and the L1 formula for the
approximation of the time fractional derivative, the fully discrete scheme for solving the
constant coefﬁcient fractional diffusion-wave equation is obtained and the superconvergence
estimate is proposed and analyzed. Further, a fully discrete ﬁnite element scheme is presented
for solving the variable coefﬁcient fractional diffusion-wave equation and the corresponding
error estimates are also established. Finally, numerical experiments are included to support
the theoretical results.
The research is supported by the Major State Research Development Program of China (No.
2016YFB0201304), National Magnetic Conﬁnement Fusion Science Program of China (No.
2015GB110003), National Natural Science Foundation of China (Nos. 11601119, 11471329, 11526074,
91430216 and U1530401), Youth Innovation Promotion Association of CAS, the Science and Technology
Program of Henan (No. 162102410003) and Foundation of Henan Educational Committee (No. 17A110002).
College of Mathematics and Information Science, Henan University of Economics and Law,
Zhengzhou 450045, China
LSEC and Institute of Computational Mathematics, Academy of Mathematics and System Science,
Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190,
Beijing Computational Science Research Center, Beijing 100094, China