A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations

Zhichao Fang; Ruixia Du; Hong Li; Yang Liu; Zhichao Fang; Ruixia Du; Hong Li; Yang Liu

doi:10.3934/math.2022112

AIMS Mathematics

2022, Volume 7, Issue 2: 1941-1970. doi: 10.3934/math.2022112

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Research article

A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Received: 11 September 2021 Accepted: 25 October 2021 Published: 04 November 2021
MSC : 65M08, 65M12, 65M15

In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.
- two-grid mixed finite volume element algorithm,
- time fractional reaction-diffusion equations,
- $ L1 $-formula,
- Browder fixed point theorem,
- error estimate
Citation: Zhichao Fang, Ruixia Du, Hong Li, Yang Liu. A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations[J]. AIMS Mathematics, 2022, 7(2): 1941-1970. doi: 10.3934/math.2022112

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Abstract

In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.

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