A high-order stabilizer-free weak Galerkin finite element method on nonuniform time meshes for subdiffusion problems

Şuayip Toprakseven; Seza Dinibutun; Şuayip Toprakseven; Seza Dinibutun

doi:10.3934/math.20231588

AIMS Mathematics

2023, Volume 8, Issue 12: 31022-31049. doi: 10.3934/math.20231588

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A high-order stabilizer-free weak Galerkin finite element method on nonuniform time meshes for subdiffusion problems

Şuayip Toprakseven ^{1
,
,},
Seza Dinibutun ²

1.
Artvin Vocational School, Accounting and Taxation, Artvin Çoruh University, Artvin, 08100, Turkey
2.
International University of Kuwait, Ardiya, Kuwait Department of Mathematics and Natural Sciences, Kuwait

Received: 18 September 2023 Revised: 09 November 2023 Accepted: 15 November 2023 Published: 20 November 2023
MSC : 65M06, 65M12, 65M60, 65M70

We present a stabilizer-free weak Galerkin finite element method (SFWG-FEM) with polynomial reduction on a quasi-uniform mesh in space and Alikhanov's higher order L2-$ 1_\sigma $ scheme for discretization of the Caputo fractional derivative in time on suitable graded meshes for solving time-fractional subdiffusion equations. Typical solutions of such problems have a singularity at the starting point since the integer-order temporal derivatives of the solution blow up at the initial point. Optimal error bounds in $ H^1 $ norm and $ L^2 $ norm are proven for the semi-discrete numerical scheme. Furthermore, we have obtained the values of user-chosen mesh grading constant $ r $, which gives the optimal convergence rate in time for the fully discrete scheme. The optimal rate of convergence of order $ \mathcal{O}(h^{k+1}+M^{-2}) $ in the $ L^\infty(L^2) $-norm has been established. We give several numerical examples to confirm the theory presented in this work.
- sub-diffusion problems,
- graded temporal meshes,
- stabilizer-free weak Galerkin finite element method,
- optimal rate of convergence
Citation: Şuayip Toprakseven, Seza Dinibutun. A high-order stabilizer-free weak Galerkin finite element method on nonuniform time meshes for subdiffusion problems[J]. AIMS Mathematics, 2023, 8(12): 31022-31049. doi: 10.3934/math.20231588

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Abstract

We present a stabilizer-free weak Galerkin finite element method (SFWG-FEM) with polynomial reduction on a quasi-uniform mesh in space and Alikhanov's higher order L2-$ 1_\sigma $ scheme for discretization of the Caputo fractional derivative in time on suitable graded meshes for solving time-fractional subdiffusion equations. Typical solutions of such problems have a singularity at the starting point since the integer-order temporal derivatives of the solution blow up at the initial point. Optimal error bounds in $ H^1 $ norm and $ L^2 $ norm are proven for the semi-discrete numerical scheme. Furthermore, we have obtained the values of user-chosen mesh grading constant $ r $, which gives the optimal convergence rate in time for the fully discrete scheme. The optimal rate of convergence of order $ \mathcal{O}(h^{k+1}+M^{-2}) $ in the $ L^\infty(L^2) $-norm has been established. We give several numerical examples to confirm the theory presented in this work.

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