A GALERKIN FORMULATION OF THE BOUNDARY ELEMENT METHOD FOR 2-DIMENSIONAL AND AXISYMMETRICAL PROBLEMS IN ELECTROSTATICS

Charge simulation and boundary element techniques typically solve for discretized charge densities on or within domain boundaries by satisfying, in general, the Cauchy condition for a discrete number of collocation points. No constraint is imposed upon the approximation except at these locations, and the boundary conditions may not be met at other points along the boundary. We 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 us 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. We describe how the integrals are evaluated for both the standard and Galerkin Boundary element functions using zero, first, and second order interpolation functions. We 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.