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

2018 ◽  
Vol 56 (3) ◽  
pp. 1482-1497 ◽  
Author(s):  
Runchang Lin ◽  
Xiu Ye ◽  
Shangyou Zhang ◽  
Peng Zhu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Singularly Perturbed ◽  
Diffusion Reaction ◽  
Galerkin Finite Element Method ◽  
Weak Galerkin ◽  
Galerkin Finite Element ◽  
Convection Diffusion ◽  
Element Method

scholarly journals Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems

2021 ◽  
Author(s):  
şuayip toprakseven ◽  
Peng Zhu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Reaction ◽  
Stability And Convergence ◽  
Optimal Order ◽  
Galerkin Finite Element Method ◽  
Weak Galerkin ◽  
Order Error ◽  
Galerkin Finite Element ◽  
Element Method

In this paper, a modified weak Galerkin finite element method on Shishkin mesh has been developed and analyzed for the singularly perturbed convection-diffusion-reaction problems. The proposed method is based on the idea of replacing the standard gradient (derivative) and convection derivative by modified weak gradient (derivative) and modified weak convection derivative, respectively, over piecewise polynomials of degree $k\geq1$. The present method is parameter-free and has less degree of freedom compared to the weak Galerkin finite element method. Stability and convergence rate of $\mathcal {O}((N^{-1}\ln N)^k)$ in the energy norm are proved. The method is uniformly convergent, i.e., the results hold uniformly regardless of the value of the perturbation parameter. Numerical experiments confirm these theoretical findings on Shishkin meshes. The numerical examples are also carried out on B-S meshes to confirm the theoretical results. Moreover, the proposed method has the optimal order error estimates of $\mathcal {O}(N^{-(k+1)})$ in a discrete $L^2-$ norm and converges at superconvergence order of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L_\infty-$ norm.

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