A Galerkin formulation of the boundary element method for two-dimensional and axi-symmetric problems in electrostatics

The authors propose to process the Fredholm integral equation relating potential to an unknown source density function by the Galerkin weighted residual technique. In essence, this allows them to optimally satisfy the Dirichlet condition over the entire conductor surface. Solving the resulting equations requires evaluation of a second surface integration over weakly singular kernels, and the increased accuracy comes at some computational expense. The singularity issue is addressed analytically for 2-D problems and semi-analytically for axi-symmetric problems. The authors describe how the integrals are evaluated for both the standard and Galerkin boundary element functions using zero, first, and second order interpolation functions. They demonstrate that the Galerkin solution is superior to the standard collocation procedure for some canonical problems, including one in which analytical charge density becomes singular