Boundary element computation and convergence analysis of electrostatic fields in a three-ring electrostatic lens

Accurate open-domain electrostatic-field computation is essential for precision charged-particle optics, but axisymmetric multi-ring electrodes introduce edge singularities and a logarithmic kernel singularity after azimuthal integration. This study develops a singularity-stabilized two-dimensional axisymmetric indirect BEM for a three-ring electrostatic lens. The formulation combines analytical azimuthal reduction to a complete-elliptic-integral kernel, local graded boundary discretization near electrode edges, and semi-analytical extraction and add-back of the coincident-element logarithmic singularity. Mesh and quadrature effects are assessed separately using Richardson-extrapolated reference solutions, Gauss-quadrature tests, and a two-parameter error-cost map. The results show nearly second-order spatial convergence and rapid decay of quadrature error to a low-level plateau. Auxiliary three-dimensional BEM and two-dimensional axisymmetric FEM calculations provide cross-validation. For the present model, the proposed BEM gives lower axial-potential errors with fewer unknowns and a shorter assembly-and-solution time than FEM. A Pareto-optimal front guides the selection of element number N and Gauss order G under accuracy and CPU-time constraints. The method provides an efficient, error-quantified basis for subsequent trajectory and aberration analyses of axisymmetric electrostatic lenses.