Applicability of the T-matrix method based on intrinsic singularities

We numerically verify geometric criteria for convergence of the extended boundary condition method (EBCM), for a range of rotationally symmetric shapes in the electrostatic limit. The theory is based off the intrinsic singularities, or foci, of the corresponding 2D curve, that lie inside and outside the shape. The convergence criteria is that the foci inside the curve must lie closer to the origin than those outside. We analyze convergence with matrix size and check numerical sources of error from oscillatory integrands, rounding errors and matrix inversion. We find that for the boundary case of discontinuous objects such as cylinders, where the internal and external foci coincide at sharp corners, the EBCM can converge. We find an example of a smooth convex boundary which fails the convergence criteria, and also investigate spheroidal harmonics as a basis where the criteria for convergence changes, allowing convergence where the spherical basis diverges. • Numerical analysis of the geometric criteria of the EBCM to converge for a boundary. • Link to foci of analytic curves. • Example of a smooth convex boundary for which the EBCM is invalid.