A weak Galerkin finite element approximation of two-dimensional sub-diffusion equation with time-fractional derivative

Ailing Zhu; Yixin Wang; Qiang Xu; Ailing Zhu; Yixin Wang; Qiang Xu

doi:10.3934/math.2020274

AIMS Mathematics

2020, Volume 5, Issue 5: 4297-4310. doi: 10.3934/math.2020274

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Research article

A weak Galerkin finite element approximation of two-dimensional sub-diffusion equation with time-fractional derivative

School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong, 250014, P. R. China

Received: 27 February 2020 Accepted: 15 April 2020 Published: 08 May 2020
MSC : 65L60, 65L70

We develop a fully discrete weak Galerkin finite element method for the initial-boundary value problem of two-dimensional sub-diffusion equation with Caputo time-fractional derivative. A traditional $L_1$ discretization for the Caputo time-fractional derivative and a weak Galerkin scheme for the space integer differential operator are employed. We prove the stability of the numerical method and establish the error estimate in $L^2$ and discrete $H^1$ norms, respectively. Some numerical results are reported to confirm the theory.
- sub-diffusion equation,
- Caputo fractional derivative,
- weak Galerkin finite element method,
- discrete weak gradient,
- error estimate
Citation: Ailing Zhu, Yixin Wang, Qiang Xu. A weak Galerkin finite element approximation of two-dimensional sub-diffusion equation with time-fractional derivative[J]. AIMS Mathematics, 2020, 5(5): 4297-4310. doi: 10.3934/math.2020274

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Abstract

We develop a fully discrete weak Galerkin finite element method for the initial-boundary value problem of two-dimensional sub-diffusion equation with Caputo time-fractional derivative. A traditional $L_1$ discretization for the Caputo time-fractional derivative and a weak Galerkin scheme for the space integer differential operator are employed. We prove the stability of the numerical method and establish the error estimate in $L^2$ and discrete $H^1$ norms, respectively. Some numerical results are reported to confirm the theory.

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