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

Houchao Zhang & Dongyang Shi

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(h2) of EQrot1 element (see Lemma 2.3). Then, by using the proved character of EQrot1 element, we present the superconvergent estimates for the original variable u in the broken H1-norm and the flux $\vec{q}$ = ∇u in the (L2)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(h2) of EQrot1 element (see Lemma 2.3). Then, by using the proved character of EQrot1 element, we present the superconvergent estimates for the original variable u in the broken H1-norm and the flux $\vec{q}$ = ∇u in the (L2)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://dor.org/10.4208/jcm.1805-m2017-0256 UR - https://global-sci.org/intro/article_detail/jcm/13005.html KW - Nonconforming MFEM KW - L1 method KW - Time-fractional diffusion equations KW - 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(h2) of EQrot1 element (see Lemma 2.3). Then, by using the proved character of EQrot1 element, we present the superconvergent estimates for the original variable u in the broken H1-norm and the flux $\vec{q}$ = ∇u in the (L2)2-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.

Houchao Zhang & Dongyang Shi. (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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