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Volume 24, Issue 1
Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

Dongfang Li, Hong-Lin Liao, Weiwei Sun, Jilu Wang & Jiwei Zhang

DOI: 10.4208/cicp.OA-2017-0080

Commun. Comput. Phys., 24 (2018), pp. 86-103.

Published online: 2018-03

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  • Abstract

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods. The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality. In this paper, we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative. In terms of the Gronwall type inequality, we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems. The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

  • Keywords

Time-fractional nonlinear parabolic problems, L1-Galerkin FEMs, Error estimates, discrete fractional Gronwall type inequality, Linearized schemes.

  • AMS Subject Headings

65M06, 35B65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-24-86, author = {Dongfang Li, Hong-Lin Liao, Weiwei Sun, Jilu Wang and Jiwei Zhang}, title = {Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems}, journal = {Communications in Computational Physics}, year = {2018}, volume = {24}, number = {1}, pages = {86--103}, abstract = {

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods. The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality. In this paper, we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative. In terms of the Gronwall type inequality, we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems. The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2017-0080}, url = {http://global-sci.org/intro/article_detail/cicp/10929.html} }
TY - JOUR T1 - Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems AU - Dongfang Li, Hong-Lin Liao, Weiwei Sun, Jilu Wang & Jiwei Zhang JO - Communications in Computational Physics VL - 1 SP - 86 EP - 103 PY - 2018 DA - 2018/03 SN - 24 DO - http://doi.org/10.4208/cicp.OA-2017-0080 UR - https://global-sci.org/intro/article_detail/cicp/10929.html KW - Time-fractional nonlinear parabolic problems, L1-Galerkin FEMs, Error estimates, discrete fractional Gronwall type inequality, Linearized schemes. AB -

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods. The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality. In this paper, we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative. In terms of the Gronwall type inequality, we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems. The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

Dongfang Li, Hong-Lin Liao, Weiwei Sun, Jilu Wang and Jiwei Zhang. (2018). Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems. Communications in Computational Physics. 24 (1). 86-103. doi:10.4208/cicp.OA-2017-0080
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