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# A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

1 College of Finance & Information, Ningbo University of Finance & Economics, Ningbo, 315175, China

2 Faculty of Science, Ningbo University of Technology, Ningbo, 315016, China

* Corresponding Author: Fengxin Sun. Email:

(This article belongs to the Special Issue: Numerical Methods in Engineering Analysis, Data Analysis and Artificial Intelligence)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(1), 341-356. https://doi.org/10.32604/cmes.2022.023140

**Received** 11 April 2022; **Accepted** 17 May 2022; **Issue published** 29 September 2022

## Abstract

By introducing the dimensional splitting (DS) method into the multiscale interpolating element-free Galerkin (VMIEFG) method, a dimension-splitting multiscale interpolating element-free Galerkin (DS-VMIEFG) method is proposed for three-dimensional (3D) singular perturbed convection-diffusion (SPCD) problems. In the DS-VMIEFG method, the 3D problem is decomposed into a series of 2D problems by the DS method, and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method. The improved interpolation-type moving least squares (IIMLS) method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems. The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems. The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes. For extremely small singular diffusion coefficients, the numerical solution will avoid numerical oscillation and has high computational stability.## Keywords

In view of the advantage that the construction of the approximation function is only related to the discrete point and not related to the grid, the meshless method has received a lot of attention from many scholars in recent years [1–4]. Meshless methods have been widely used in scientific and engineering problems and have shown high accuracy and effectiveness [5–9].

Based on various construction methods of approximation functions or various discrete methods of the problem to be solved, many meshless methods have been proposed [10–14]. The ordinary least-squares (OLS) method is the best approximation [15], and it has been applied in engineering fields widely [16,17]. Based on the OLS method, Lancaster et al. presented the moving least-squares method (MLS) approximation [18], which is one of the common methods used to construct approximation functions and has a wide range of applications in meshless [18]. Many meshless methods have been proposed based on the MLS method, such as element-free Galerkin method [19], smoothed particle hydrodynamics [20] and meshless local Petrov-Galerkin method [21]. From the research of many scholars, the meshless method based on MLS has better effectiveness [22]. Based on MLS, some improved methods have also been proposed, such as complex variable moving least squares method [23,24], interpolation-type moving least-squares (IMLS) method [25]. To avoid the singularity of the weight function, Cheng et al. [26–28] also proposed an improved interpolation-type moving least squares (IIMLS) method with non-singular weights.

The element-free Galerkin (EFG) method, which couples the Galerkin weak form and the MLS method, is a widely used mesh-free method [29]. In order to directly apply essential boundary conditions, the interpolating element-free Galerkin (IEFG) method is proposed by coupling the IMLS method [28,30–33]. The IEFG method not only has the advantage of directly applying boundary conditions but also has the advantage of having a smaller radius of influence compared to EFG under the same basic functions. The IEFG method has been applied to potential problems [34,35], elastoplasticity problems [28], crack problems [36], structural dynamic analysis [37], prevention of groundwater contamination [38], elastoplasticity problems [39], Poisson equation [40], elastic large deformation problems [41], Oldroyd equation [42], etc.

To improve the computational efficiency of the EFG method, by introducing the dimension splitting method [43], Cheng et al. proposed the dimension splitting element-free Galerkin (DS-EFG) method [44] and dimension splitting interpolating element-free Galerkin (DS-IEFG) method [45]. The dimension splitting meshless method greatly improves the computational efficiency of the EFG method, and shows high computational efficiency and accuracy for 3D advection-diffusion problems [46], 3D transient heat conduction problems [47–49], 3D elasticity problems [50], 3D wave equations [51,52], etc.

For some fluid problems with large Reynolds numbers, the solution of EFG method may have non-physical oscillations. In order to avoid the physical oscillation, Ouyang et al. [53] proposed the variational multiscale element-free Galerkin (VMEFG) method by introducing variational multiscale (VM) method. The VMEFG method has high stability for fluid problems with large Reynolds numbers or singular disturbances [2,54,55]. Similar to the EFG method, the DS-EFG method is also prone to nonphysical oscillations for singularly perturbed fluid problems. By coupling the VM and DS-EFG methods, Wang et al. presented the hybrid variational multiscale element-free Galerkin method for 2D convection-diffusion [4,56] and 2D Stokes problems [57].

The convection-diffusion (CD) equation plays an important role in some physical problems [58,59], such as the transport of the quantity in air and river pollution. Since it is often difficult to obtain analytical solutions, many scholars have studied numerical methods to obtain approximate solutions. Numerical instability is prone to occur when the CD problem contains large Reynolds numbers or singularly perturbed diffusion coefficients [4]. Stabilization techniques must be added to numerical methods to avoid numerical oscillations of the solution. Varying the shape parameter, a finite-difference method based on the radial basis function is introduced for 2D steady CD equations with large Reynolds’ numbers [60]. Aga presented an improved finite difference method for 1D singularly perturbed CD problems [61]. Using the collocation method of high-order polynomial approximation, Ömer studied the numerical solution of 3D CD problems with high Reynolds [62]. Zhang et al. [63] presented a VMIEFG method for the singularly perturbed two-dimensional CD problems.

In this paper, by introducing the dimension splitting (DS) method into the VMIEFG method, we will develop a dimension splitting multiscale interpolating element-free Galerkin (DS-VMIEFG) method for three-dimensional singular perturbed convection-diffusion (SPCD) problems. In the DS-VMIEFG method, the DS method is used to decompose the 3D problem into a series of 2D problems, and the discrete equations on the 2D splitting surface will be obtained by the VMIEFG method. The IIMLS method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems. And some numerical examples will be solved to verify the effectiveness of the DS-VMIEFG method.

In order to overcome the insufficiency that the weight function must be singular in the IMLS method, by coupling the interpolation function transformation and MLS approximation, Cheng et al. [26] presented the IIMLS method. In this method, the approximation function satisfies the interpolation property, and the weight function is also non-singular.

Let

where

with

The element in the i-th row and j-th column of matrix

3 The DS-VMIEFG Method for 3D SPCD Problems

The following stationary three-dimensional convection-diffusion problem is considered:

where

By using the dimension splitting method, the problem in (12) can be transformed into a series of 2D problems on the coordinate plane of

where

Define the subscript notation in partial derivatives as

The variational weak form of Eq. (13) is

where

Using the variational multiscale method, the functions are broken down into two parts of the coarse and fine scales as

From Eqs. (18) and (16), we have

Decomposing Eq. (19) into coarse-scale and fine-scale parts leads to

and

Suppose

where

and

From Eq. (21), it follows that

where

If let

where

On the boundary, the fine-scale function can be seen as zero. The coefficient

where

Then it follows that

where

Omitting the superposed bars and coupling Eqs. (20) and (33), we have

where

The summation expression in Eq. (34) denotes the effect of the fine scale for obtaining stale solutions.

On

where

The Galerkin weak form of Eq. (34) is: for

where

Omitting the higher derivatives in the stability term, from Eq. (37), we can obtain the linear equations as

where

Eq. (38) is the discrete system of equations on the 2D dimension-splitting surface

where

Using Eqs. (46) and (47) and the interpolating property, the whole discrete equations of the DS-VMIEFG method for the 3D SPCD problems can be assembled as

Substituting the boundary condition into Eq. (48) directly, the solutions for the three-dimensional convection-diffusion problem will be solved from Eq. (48).

In this section, the validity of the method of this paper will be verified by two examples. We take the cubic spline function as the weight function in the IIMLS method. The integration scheme uses a rectangular

Define the relative error by

where

Example 1. The first consideration is a singularly perturbed convection-diffusion problem on a cube with an exact solution as

The velocity field parameters are fixed to be

When the small coefficients are respectively

When the nodes distribution is

To study the convergence, when there are 21 splitting points in the

Example 2. The second considered convection-diffusion problem has the following exact solution as [62]

The parameters are

When

When

When using the regular

When 21 splitting points are fixed in the z direction, for different node distributions of

By introducing the DS method into the VMIEFG method, a DS-VMIEFG method for three-dimensional singular perturbed convection-diffusion problems is presented in this paper. In the DS-VMIEFG method, the 3D problem is decomposed into a series of 2D problems, and then the weak form of the Galerkin integral is only established on the 2D splitting surfaces by the VMIEFG method. The DS-VMIEFG method can avoid the construction of integral weak form on the 3D domain. The IIMLS method is used to obtain the shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the 3D SPCD problems. The numerical example verifies the effectiveness of the DS-VMIEFG method in the case of very small singularly perturbed diffusion coefficients, and the numerical solution can avoid non-physical numerical oscillations.

Funding Statement: This work is supported by the Natural Science Foundation of Zhejiang Province, China (Grant Nos. LY20A010021, LY19A010002, LY20G030025), and the Natural Science Foundation of Ningbo City, China (Grant Nos. 2021J147, 2021J235).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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*135*

*(1)*, 341-356. https://doi.org/10.32604/cmes.2022.023140

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*Comput. Model. Eng. Sci.*, vol. 135, no. 1, pp. 341-356. 2023. https://doi.org/10.32604/cmes.2022.023140