Superconvergence of a modified weak Galerkin method for singularly perturbed two-point elliptic boundary-value problems

Toprakseven, Şuayip

doi:10.1007/s10092-021-00449-y

Cited by 11 publications

(3 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

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

View full text Add to dashboard Cite

show abstract

“…MWG-FEMs have been further developed for a variety of PDEs such as convection-diffusion problems [12], parabolic equations [11], Stokes equations [20,26], convection-diffusion problems in one dimension [28] and in higher dimension with weakly imposed boundary condition [9]. Superconvergence approximation of the MWG-FEM is also presented in [29]. This paper aims to introduce a MWG-FEM for time dependent convection-diffusion problems.…”

Section: Introductionmentioning

confidence: 99%

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

Gao¹

2023

JCM

View full text Add to dashboard Cite

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.

show abstract

“…The WG method has been applied to all kinds of problems including Stokes equations [26,27,29], Maxwell's equations [16], Brinkman equations [15,28,31], fractional time convection-diffusion problems [21] and so on. For singular perturbed value problems, the WG method has also yielded some results, such as [9,23,24,22,34,36,33,3]. The main purpose of this paper is to present optimal order uniform convergence in the energy norm on Bakhvalov-type mesh for convection-dominated problems in 2D.…”

Section: Introductionmentioning

confidence: 99%

Supercloseness of linear streamline diffusion finite element method on Bakhvalov-type mesh for singularly perturbed convection-diffusion equation in 1D

Liu

Zhang

2022

Applied Mathematics and Computation

View full text Add to dashboard Cite

Superconvergence of a modified weak Galerkin method for singularly perturbed two-point elliptic boundary-value problems

Cited by 11 publications

References 30 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

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

Supercloseness of linear streamline diffusion finite element method on Bakhvalov-type mesh for singularly perturbed convection-diffusion equation in 1D

Contact Info

Product

Resources

About