On the pseudo-differential operators in the dual boundary integral equations using degenerate kernels and circulants

Abstract

The spectral properties for the six kernels (influence matrices) in the dual boundary integral equations (dual BEM) are investigated for the Laplace and Helmholtz equations of a circular domain. Based on the two-point functions for the six kernels of single layer, double layer, normal derivatives of single and double layer potentials, tangent derivatives of single and double layer potentials, they can be expressed in degenerate kernels. Using the analytical properties of circulants, the spectral properties are studied exactly in a discrete system for a circular cavity when a uniform constant element scheme is adopted. After considering the number of degrees of freedom for the discrete system to be infinite for continuous system, the spectral properties of continuous system can be obtained. The relation for the influence matrices between the interior and exterior problems is addressed. Also, the condition number for the matrices and the orders of the pseudo-differential operators are examined. Finally, the properties of Calderon projector in discrete formulation are derived and are demonstrated analytically by an example of circular domain. Also, numerical results using the dual BEM program are performed to check the identities for the Calderon projector.