A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients

Toprakseven, Şuayip

doi:10.1016/j.apnum.2021.05.021

Cited by 28 publications

(5 citation statements)

References 34 publications

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“…The key feature of this method is that the classical derivative is replaced by weak derivative in the corresponding variational formulation in a way that completely discontinuous functions have been allowed to use in the numerical scheme which has a parameter independent stabilizer. The weak Galerkin method has been studied and applied to a variety of problems including Stokes equations [32], interface problem [18], Maxwell equation [19], fractional time convection-diffusion problems [28], and singularly perturbed elliptic equations in one and higher dimensions [10,29,30].…”

Section: Introductionmentioning

confidence: 99%

Optimal order uniform convergence of weak Galerkin finite element method on Bakhvalov-type meshes for singularly perturbed convection dominated problems

Toprakseven

2023

Hacettepe Journal of Mathematics and Statistics

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In this paper, we propose a weak Galerkin finite element method (WG-FEM) for solving two-point boundary value problems of convection-dominated type on a Bakhvalov-type mesh. A special interpolation operator which has a simple representation and can be easily extended to higher dimensions is introduced for convection-dominated problems. A robust optimal order of uniform convergence has been proved in the energy norm with this special interpolation using piecewise polynomials of degree $k\geq 1$ on interior of the elements and piecewise constant on the boundary of each element. The proposed finite element scheme is independent of parameter and since the interior degree of freedom can be eliminated efficiently from the resulting discrete system, number of unknowns of the proposed method is comparable with the standard finite element methods. An optimal order of uniform convergence is derived on Bakhvalov-type mesh. Finally, numerical experiments are given to support the theoretical findings and show the efficiency of the proposed method.

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Section: Introductionmentioning

confidence: 99%

Optimal order uniform convergence of weak Galerkin finite element method on Bakhvalov-type meshes for singularly perturbed convection dominated problems

Toprakseven

2023

Hacettepe Journal of Mathematics and Statistics

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“…They pointed out that their schemes are well-suited to time FPDEs since they preserve the singularity of the solution. Toprakseven ( 2021 ) utilized the classical L1 discretization in time and a weak Galerkin element in space to establish a weak Galerkin finite element method for the d -dimensional ( ) TFADRE. Kumar and Zeidan ( 2021 ) scrutinized the one-dimensional non-linear TFADRE in which the fractional temporal derivative is defined in the Atangana-Baleanu sense.…”

Section: Introductionmentioning

confidence: 99%

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

et al. 2023

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The time-fractional advection–diffusion reaction equation (TFADRE) is a fundamental mathematical model because of its key role in describing various processes such as oil reservoir simulations, COVID-19 transmission, mass and energy transport, and global weather production. One of the prominent issues with time fractional differential equations is the design of efficient and stable computational schemes for fast and accurate numerical simulations. We construct in this paper, a simple and yet efficient modified fractional explicit group method (MFEGM) for solving the two-dimensional TFADRE with suitable initial and boundary conditions. The proposed method is established using a difference scheme based on L1 discretization in temporal direction and central difference approximations with double spacing in spatial direction. For comparison purposes, the Crank–Nicolson finite difference method (CNFDM) is proposed. The stability and convergence of the presented methods are theoretically proved and numerically affirmed. We illustrate the computational efficiency of the MFEGM by comparing it to the CNFDM for four numerical examples including fractional diffusion and fractional advection–diffusion models. The numerical results show that the MFEGM is capable of reducing iteration count and CPU timing effectively compared to the CNFDM, making it well-suited to time fractional diffusion equations.

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“…The weak functions in WG-FEMs consist of the form u = {u 0 , u b } with u = u 0 inside of the element and u = u b on the boundary of the element. Later on, WG finite element methods have further been presented for a large variety of PDEs including the implementation results [21], parabolic problems [16], the Helmholtz equations with high wave numbers in [22] and the time fractional reaction-diffusion-convection problems in [27]. The weak gradient and weak divergence operators have been introduced for convection-dominated problems in [5] and [17].…”

Section: Introductionmentioning

confidence: 99%

A Modified Weak Galerkin Finite Element Method for Singularly Perturbed Parabolic Convection-Diffusion-Reaction Problems

Gao¹

2023

JCM

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In this work, a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations. The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators, respectively. We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh. The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods. Optimal order of convergences are obtained in suitable norms. We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method. Various numerical examples are presented to support the theoretical results. It is theoretically and numerically shown that the method is quite stable.

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A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients

Cited by 28 publications

References 34 publications

Optimal order uniform convergence of weak Galerkin finite element method on Bakhvalov-type meshes for singularly perturbed convection dominated problems

Optimal order uniform convergence of weak Galerkin finite element method on Bakhvalov-type meshes for singularly perturbed convection dominated problems

Efficient numerical simulations based on an explicit group approach for the time fractional advection–diffusion reaction equation

A Modified Weak Galerkin Finite Element Method for Singularly Perturbed Parabolic Convection-Diffusion-Reaction Problems

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