High-accuracy quadrature methods for solving boundary integral equations of steady-state anisotropic heat conduction problems with Dirichlet conditions

Abstract

This paper presents the mechanical quadrature methods (MQMs) for solving the boundary integral equations of steady-state anisotropic heat conduction equation on the smooth domains and polygons, respectively. The costless and high-accurate Sidi–Israeli quadrature formula are applied to deal with the integrals in which the kernels have a logarithmic singularity. Especially, the Sidi transformation is used for the polygon cases in order to obtain a rapid convergence by degrading the singularity at the corners on the boundary. The convergence and stability of the MQMs solution are proved based on Anselone's collective compact theory. In addition, asymptotic error expansion of the MQMs shows that the approximation order is of O(h³), where h is the partition size of the boundary. Finally, numerical examples are tested and results verify the theoretical analysis.

Keywords:

• anisotropic heat conduction
• mechanical quadrature methods
• singularity
• boundary integral equation
• condition number

2000 AMS Subject Classifications:

• 65N38
• 65R20

Acknowledgements

The work was supported by the National Natural Science Foundation of China (10871034).