Analytical integration of elliptic 2D fundamental solution and its derivatives for straight-line elements with constant interpolation

摘要

This article describes the analytical integration of the elliptic 2D fundamental solution and its lst, 2nd and 2nd derivatives with the constant function interpolation for straight-line boundary elements. As a result of the character of the integrated function, the integrals are characterized with regard to position of the source point. It it lies on the boundary. Г, then the integrals are: weak-(log r), strong-(1/r) and hyer-(1/r~2) singular. Otherwise, the integrals are regular. The 3rd derivatives of the fundamental solution are needed for the calculation of а~2u/аx~2 and а~2u/а y~2 of harmonic function u in the domain Ω. A comparison of the analytical and the numerical integrations is made.