Variational multiscale element free Galerkin method for convection-diffusion-reaction equation with small diffusion

2014 ◽  
Vol 46 ◽  
pp. 85-92 ◽  
Author(s):  
Xiaohua Zhang ◽  
Hui Xiang
Keyword(s):  
Galerkin Method ◽  
Diffusion Reaction ◽  
Element Free Galerkin Method ◽  
Reaction Equation ◽  
Small Diffusion ◽  
Convection Diffusion ◽  
Variational Multiscale ◽  
Element Free Galerkin ◽  
Element Free

scholarly journals Variational Multiscale Element Free Galerkin Method Coupled with Low-Pass Filter for Burgers’ Equation with Small Diffusion

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Ping Zhang ◽  
Xiaohua Zhang ◽  
Laizhong Song
Keyword(s):  
Burgers Equation ◽  
Element Free Galerkin Method ◽  
Pass Filter ◽  
Small Diffusion ◽  
Low Pass Filter ◽  
Variational Multiscale ◽  
Element Free Galerkin ◽  
Low Pass ◽  
Interior Layers ◽  
Element Free

Variational multiscale element free Galerkin (VMEFG) method is applied to Burgers’ equation. It can be found that, for the very small diffusivity coefficients, VMEFG method still suffers from instability in the presence of boundary or interior layers. In order to overcome this problem, the high order low-pass filter is used to smooth the solution. Three test examples with very small diffusion are presented and the solutions obtained are compared with exact solutions and some other numerical methods. The numerical results are found in which the VMEFG coupled with low-pass filter works very well for Burgers’ equation with very small diffusivity coefficients.

scholarly journals A Dimension Splitting Generalized Interpolating Element-Free Galerkin Method for the Singularly Perturbed Steady Convection–Diffusion–Reaction Problems

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2524
Author(s):  
Fengxin Sun ◽  
Jufeng Wang ◽  
Xiang Kong ◽  
Rongjun Cheng
Keyword(s):  
Numerical Solutions ◽  
Splitting Method ◽  
Singularly Perturbed ◽  
Diffusion Reaction ◽  
Element Free Galerkin Method ◽  
Convection Diffusion ◽  
Discrete Equations ◽  
Element Free Galerkin ◽  
Element Free ◽  
Steady Convection

By introducing the dimension splitting method (DSM) into the generalized element-free Galerkin (GEFG) method, a dimension splitting generalized interpolating element-free Galerkin (DS-GIEFG) method is presented for analyzing the numerical solutions of the singularly perturbed steady convection–diffusion–reaction (CDR) problems. In the DS-GIEFG method, the DSM is used to divide the two-dimensional CDR problem into a series of lower-dimensional problems. The GEFG and the improved interpolated moving least squares (IIMLS) methods are used to obtain the discrete equations on the subdivision plane. Finally, the IIMLS method is applied to assemble the discrete equations of the entire problem. Some examples are solved to verify the effectiveness of the DS-GIEFG method. The numerical results show that the numerical solution converges to the analytical solution with the decrease in node spacing, and the DS-GIEFG method has high computational efficiency and accuracy.

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