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.

A Hybrid Variational Multiscale Element-Free Galerkin Method for Convection-Diffusion Problems

2019 ◽  
Vol 11 (07) ◽  
pp. 1950063 ◽  
Author(s):  
Jufeng Wang ◽  
Fengxin Sun
Keyword(s):  
Splitting Method ◽  
Two Dimensional ◽  
One Dimensional ◽  
Convection Diffusion ◽  
Variational Multiscale ◽  
Discrete Equations ◽  
Element Free Galerkin ◽  
Oscillating Solutions ◽  
Element Free ◽  
Diffusion Problems

By coupling the dimension splitting method (DSM) and the variational multiscale element-free Galerkin (VMEFG) method, a hybrid variational multiscale element-free Galerkin (HVMEFG) method is developed for the two-dimensional convection-diffusion problems. In the HVMEFG method, the two-dimensional problem is converted into a battery of one-dimensional problems by the DSM. Combining the non-singular improved interpolating moving least-squares (IIMLS) method, the VMEFG method is used to obtain the discrete equations of the one-dimensional problems on the splitting plane. Then, final discretized equations of the entire convection-diffusion problems are assembled by the IIMLS method. The HVMEFG method has high accuracy and efficiency. Numerical examples show that the HVMEFG method can obtain non-oscillating solutions and has higher efficiency and accuracy than the EFG and VMEFG methods for convection-diffusion problems.

An Improved Element-Free Galerkin Method for a Kind of KdV Equations

2011 ◽  
Vol 101-102 ◽  
pp. 471-474
Author(s):  
Feng Xin Sun ◽  
Ju Feng Wang
Keyword(s):  
Galerkin Method ◽  
Kdv Equation ◽  
Delta Function ◽  
Element Free Galerkin Method ◽  
Essential Boundary Condition ◽  
Discrete Equations ◽  
Element Free Galerkin ◽  
Numerical Effort ◽  
The Galerkin Method ◽  
Element Free

An improved element-free Galerkin method is presented for the numerical solution of the third-order nonlinear KdV equation by coupling the interpolating moving least-squares (IMLS) method with the Galerkin method. The shape function of the IMLS method satisfies the property of Kronecker Delta function, and then the essential boundary condition can be applied directly and easily without any additional numerical effort. A variational method is used to obtain the discrete equations. A numerical example is given to demonstrate the effectiveness of the method presented in this paper for KdV equation.

Numerical Analysis for Convection-Diffusion Problems using Element-Free Galerkin Method

2000 ◽  
Vol 2000.2 (0) ◽  
pp. 5-6
Author(s):  
Changcheng SHAO ◽  
Yasuhiro MATSUDA ◽  
Naoharu Hasagawa
Keyword(s):  
Numerical Analysis ◽  
Galerkin Method ◽  
Element Free Galerkin Method ◽  
Convection Diffusion ◽  
Element Free Galerkin ◽  
Element Free ◽  
Diffusion Problems

The Element-Free Galerkin Method for the Unsteady Magnetohydrodynamic Flow in a Duct

2015 ◽  
Vol 723 ◽  
pp. 181-185
Author(s):  
Xing Hui Cai ◽  
Hong Fu Qiang ◽  
Jiang Ren Lu ◽  
Guo Liang Wang
Keyword(s):  
Galerkin Method ◽  
Numerical Solutions ◽  
Magnetohydrodynamic Flow ◽  
Least Square ◽  
Element Free Galerkin Method ◽  
Moving Least Square ◽  
Coupled Equations ◽  
Element Free Galerkin ◽  
Moving Least Square Approximation ◽  
Element Free

In this paper, a meshless global element-free Galerkin method is given to obtain the numerical solutions of the coupled equations in the velocity and magnetic field for the unsteady magnetohydrodynamic flow through a straight duct with arbitrary electrical conductivity, that is, from perfectly conducting to insulated duct walls. The moving least-square approximation scheme is employed to construct shape functions. A time stepping method is employed to deal with the time derivatives. Non-uniform background grids and nodes are applied for numerical simulations. Computations are performed for different Hartmann numbers and wall conductivities at different time.

Meshless Element-Free Galerkin Method for Ground-Coupled Heat Transfer Problems

2012 ◽  
Vol 263-266 ◽  
pp. 3292-3297
Author(s):  
Xiang Qian Li ◽  
Wei Wei Wang ◽  
Ming Hai Li ◽  
Mao Yu Zhen
Keyword(s):  
Heat Transfer ◽  
Galerkin Method ◽  
Soil Layer ◽  
Homogeneous Medium ◽  
Element Free Galerkin Method ◽  
Coupled Heat Transfer ◽  
Discrete Equations ◽  
Element Free Galerkin ◽  
Element Free ◽  
Transfer Problems

A meshless element-free Galerkin method (EFGM) which is applicable to arbitrary shapes but requires only nodal data is applied to two-dimensional steady-state ground-coupled heat transfer problems. The soil layer around underground constructions is modeled as a homogeneous medium and as a layered soil with two layers. Variational method is utilized to obtain the discrete equations. Moving least squares (MLS) approximants are used to construct the shape functions. Lagrange multiplier technique is employed to enforce the essential boundary conditions. The calculation precision of EFGM is validated by comparing EFGM results with those obtained by finite element method (FEM). EFGM reduce considerably the preparation of the model. EFGM is very appropriate for the ground-coupled heat transfer problems.

The Improved Element-Free Galerkin Method Based on the Nonsingular Weight Functions for Inhomogeneous Swelling of Polymer Gels

2018 ◽  
Vol 10 (04) ◽  
pp. 1850047 ◽  
Author(s):  
Fengbin Liu ◽  
Yumin Cheng
Keyword(s):  
Numerical Solutions ◽  
Weak Form ◽  
Polymer Gels ◽  
Weight Functions ◽  
Element Free Galerkin Method ◽  
Displacement Boundary ◽  
Element Free Galerkin ◽  
Node Distribution ◽  
Penalty Factor ◽  
Element Free

In this paper, the interpolating moving least-squares (IMLS) method based on a nonsingular weight function is used to construct the approximation function, the weak form of the problem of inhomogeneous swelling of polymer gels is used to obtain the final discretized equations, and penalty method is applied to impose the displacement boundary condition, then an improved element-free Galerkin (IEFG) method for the problem of the inhomogeneous swelling of polymer gels is presented. Three selected examples of inhomogeneous swelling of polymer gels solved with the IEFG method are given in this paper. The accuracy of the numerical solutions of the IEFG method are discussed by using different weight functions, penalty factor, scale parameter of influence domain, node distribution and step number. Numerical results of the IEFG method for inhomogeneous swelling of polymer gels show that this method has great precision, and it can solve large deformation problems of polymer gels effectively.

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