The singular boundary method (SBM) is a recent boundarytype collocation scheme with the merits of being free of mesh and integration, mathematically simple, and easytoprogram. Its essential technique is to introduce the concept of the source intensity factors to eliminate the singularities of fundamental solutions upon the coincidence of source and collocation points in a strongform formulation. In recent years, several numerical and semianalytical techniques have been proposed to determine source intensity factors. With the help of these latest techniques, this short communication makes an extensive investigation on numerical efficiency and convergence rates of the SBM to an extensive variety of benchmark problems in comparison with the BEM. We find that in most cases the SBM and BEM have similar convergence rates, while the SBM has slightly better accuracy than the direct BEM. And the condition number of SBM is lower than BEM. Without mesh and numerical integration, the SBM is computationally more efficient than the BEM.
The boundary element method (BEM) [
To remedy these drawbacks in the BEM and MFS, several numerical schemes have been proposed, such as the boundary collocation method (BCM) [
The singular boundary method (SBM) [
This short communication will make a comparison on numerical efficiency and convergence rate under the extensive benchmark testing with the best approach to determining the source intensity factors. A brief outline of the paper is as follows: Section
This section will describe the techniques to determine source intensity factors for solving 2D and 3D Laplace, Helmholtz, and modified Helmholtz equations.
This section describes the SBM for Laplace equations. Consider the Laplace equations:
By adopting the fundamental solution
Therefore, to solve all kinds of physical and mechanical problems with formulations (
In recent years, four techniques have been proposed to determine the abovementioned source intensity factors. And the merits and demerits of these techniques have already been extensively investigated in some literatures [
In this study, we use formulas (
The formulation can be expressed as
The schematic configuration of (a) the source points
Then, we use the following formula to determine the source intensity factors for 2D Laplace equations with Dirichlet boundary condition:
And we use formula (
Formulas (
In analogy to the SBM for Laplace equations, the corresponding SBM approximate solution for Helmholtz and modified Helmholtz equations can be represented as
Due to the property of the same order of the singularities between the fundamental solutions of Helmholtztype equations and Laplace equation, the source intensity factors of Helmholtz equation and modified Helmholtz equation can be expressed as
In this section, the efficiency, accuracy, and convergence of the SBM are tested to 2D and 3D Laplace, Helmholtz, and modified Helmholtz problems. The numerical accuracy is calculated by the relative root mean square errors (Rerr)
Consider 2D Laplace problems in a circular domain with radius 1. And the exact solution is given by
In this example, we use formula (
Relative errors of Example
Condition numbers of the SBM and the BEM in Example
Consider 2D Laplace problems in a square domain with length 1. And the exact solution is given by
In this example, we use formulas (
Relative errors of Example
Consider 2D Helmholtz problems in a circular domain with radius 1. And the exact solution is given by
In this example, we use formulas (
Numerical results for Helmholtz problem in circular domain.






SBM  BEM  SBM  BEM  SBM  BEM  
40 






80 






160 






320 






640 






1280 






Relative errors of Example
Consider 2D Helmholtz problems in a square domain with length 1. And the exact solution is given by
In this example, we use formulas (
Relative errors of Example
Consider 2D modified Helmholtz problems in a circular domain with radius 1. And the exact solution is given by
In this example, we use formulas (
Relative errors of Example
Consider 2D modified Helmholtz problems in a square domain with length 1. And the exact solution is given by
In this example, we use formulas (
Numerical results for Helmholtz problem in square domain.






SBM  BEM  SBM  BEM  SBM  BEM  
60 






120 






240 






480 






960 






1920 






Relative errors of Example
It can be observed that the present SBM performs more accurate solutions than the BEM with the same number of boundary nodes, and the SBM converge remarkably with the increasing boundary node number
Consider 3D Laplace problems in a sphere domain with radius 1. And the exact solution is given by
In this example, we use formula (
Relative errors of Example
Consider 3D Laplace problems in a cube domain with length 1. And the exact solution is given by
In this example, we use formulas (
Relative errors of Example
Consider 3D Laplace problems in a tire domain. The tire surface is defined by the following equation:
In this example, we use formula (
The distribution of source points on tire surface.
Relative errors of Example
Consider 3D Helmholtz problems in a sphere domain with radius 1. And the exact solution is given by
all Dirichlet boundary conditions:
all Neumann boundary conditions:
In this example, we use formulas (
Numerical results for Helmholtz problem in sphere domain in case (






SBM  BEM  SBM  BEM  SBM  BEM  
100 






400 






900 






1600 






2500 






Relative errors of Example
Relative errors of Example
Consider 3D Helmholtz problems in a cube domain with length 1 and the exact solution is given by
all Neumann boundary conditions:
In this example, we use formulas (
Numerical results for Helmholtz problems in cube domain in case (






SBM  BEM  SBM  BEM  SBM  BEM  
96 






384 






864 






1536 






2400 






Relative errors of Example
Relative errors of Example
Condition numbers of Example
Consider 3D Helmholtz problems in a tire domain. The tire surface is defined by the following equation:
all Dirichlet boundary conditions:
all Neumann boundary conditions:
In this example, we use formulas (
Relative errors of Example
Relative errors of Example
Consider 3D modified Helmholtz problems in a sphere domain with radius 1. And the exact solution is given by
all Dirichlet boundary conditions:
all Neumann boundary conditions:
In this example, we use formulas (
Relative errors of Example
Relative errors of Example
Consider 3D modified Helmholtz problems in a cube domain with length 1 and the exact solution is given by
all Neumann boundary conditions:
In this example, we use formulas (
Numerical results for modified Helmholtz problems in cube domain in case (






SBM  BEM  SBM  BEM  SBM  BEM  
96 






384 






864 






1536 






2400 






Relative errors of Example
Relative errors of Example
Consider 3D modified Helmholtz problems in a tire domain. The tire surface is defined by the following equation:
all Dirichlet boundary conditions:
all Neumann boundary conditions:
In this example, we use formulas (
It can be observed that the present SBM performs more accurate solutions than the BEM with the same number of boundary nodes, and the SBM converge remarkably with the increasing boundary node number
Relative errors of Example
Relative errors of Example
This short communication makes an extensive investigation on numerical efficiency and convergence rates of the SBM to several 2D and 3D benchmark examples about Laplace, Helmholtz, and modified Helmholtz equations in comparison with the direct BEM.
Through numerical experiments, we find that the present SBM results are in good agreement with the exact solutions in both 2D and 3D problems. And the SBM converge remarkably with the increasing boundary node number
Furthermore, we find that in most cases the SBM and BEM have similar convergence rates, while the SBM has slightly better accuracy than the direct BEM. In addition, the condition number of SBM is lower than BEM in most cases. Without mesh and integration, the SBM is computationally more efficient than the BEM.
The authors declare that they have no competing interests.
This work was supported by the National Natural Science Foundation of China (Grants nos. 11372097, 11302069, and 11572111), The National Science Fund for Distinguished Young Scholars of China (Grant no. 11125208), the 111 Project (Grant no. B12032), the Chinese Postdoctoral Science Foundation (Grant no. 2014M561565), and the Foundation for Open Project of the State Key Laboratory of Acoustics (Grant no. SKLA201509). And the authors would like to thank Dr. Linlin Sun’s help in BEM programming.