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Volume 37, Issue 4
Superconvergence Analysis for Time-Fractional Diffusion Equations with Nonconforming Mixed Finite Element Method

Houchao Zhang & Dongyang Shi

DOI: 10.4208/jcm.1805-m2017-0256

J. Comp. Math., 37 (2019), pp. 488-505.

Published online: 2019-02

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  • Abstract

In this paper, a fully discrete scheme based on the $L1$ approximation in temporal direction for the fractional derivative of order in (0, 1) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order $O(h^2)$ of $EQ^{rot}_1$ element (see Lemma 2.3). Then, by using the proved character of $EQ^{rot}_1$ element, we present the superconvergent estimates for the original variable $u$ in the broken $H^1$-norm and the flux $\vec{q} = ∇u$ in the $(L^2)^2$-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.

  • Keywords

Nonconforming MFEM, $L1$ method, Time-fractional diffusion equations, Superconvergence.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

zhc0375@126.com (Houchao Zhang)

dy_shi@zzu.edu.cn (Dongyang Shi)

  • BibTex
  • RIS
  • TXT
@Article{JCM-37-488, author = {Zhang , Houchao and Shi , Dongyang}, title = {Superconvergence Analysis for Time-Fractional Diffusion Equations with Nonconforming Mixed Finite Element Method}, journal = {Journal of Computational Mathematics}, year = {2019}, volume = {37}, number = {4}, pages = {488--505}, abstract = {

In this paper, a fully discrete scheme based on the $L1$ approximation in temporal direction for the fractional derivative of order in (0, 1) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order $O(h^2)$ of $EQ^{rot}_1$ element (see Lemma 2.3). Then, by using the proved character of $EQ^{rot}_1$ element, we present the superconvergent estimates for the original variable $u$ in the broken $H^1$-norm and the flux $\vec{q} = ∇u$ in the $(L^2)^2$-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1805-m2017-0256}, url = {http://global-sci.org/intro/article_detail/jcm/13005.html} }
TY - JOUR T1 - Superconvergence Analysis for Time-Fractional Diffusion Equations with Nonconforming Mixed Finite Element Method AU - Zhang , Houchao AU - Shi , Dongyang JO - Journal of Computational Mathematics VL - 4 SP - 488 EP - 505 PY - 2019 DA - 2019/02 SN - 37 DO - http://doi.org/10.4208/jcm.1805-m2017-0256 UR - https://global-sci.org/intro/article_detail/jcm/13005.html KW - Nonconforming MFEM, $L1$ method, Time-fractional diffusion equations, Superconvergence. AB -

In this paper, a fully discrete scheme based on the $L1$ approximation in temporal direction for the fractional derivative of order in (0, 1) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order $O(h^2)$ of $EQ^{rot}_1$ element (see Lemma 2.3). Then, by using the proved character of $EQ^{rot}_1$ element, we present the superconvergent estimates for the original variable $u$ in the broken $H^1$-norm and the flux $\vec{q} = ∇u$ in the $(L^2)^2$-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.

Zhang , Houchao and Shi , Dongyang. (2019). Superconvergence Analysis for Time-Fractional Diffusion Equations with Nonconforming Mixed Finite Element Method. Journal of Computational Mathematics. 37 (4). 488-505. doi:10.4208/jcm.1805-m2017-0256
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