Solution Convergence in Exterior Electromagnetic Boundary Value Problems and Its Dependence on the Numerical Methods-a Review

In numerical solution of electromagnetic problems, boundary conditions play an essential role to ensure the uniqueness of the solution. The problem may be solved by a variety of methods, such as the integral or differential equation methods, a modal expansion method, or some random application of boundary conditions on the object. In any case, the application of boundary conditions is used to determine the unknown field quantity of the formulation to be used for determining the entire solution in the desired space. The convergence of the solution, therefore, depends on the selection of the unknown and the method used for its determination. However, in exterior boundary value problems the field quantities of interest are at far distances from the boundaries, where the field behaviors are dependent on the entire boundary of the object. Thus, the behaviour of the far field vectors is different from those on the boundary. This problem is investigated here by means of simple scattering by conducting objects having simple, as well as, complex shapes. It is shown that, while the convergence of the solution for determining the boundary fields can be slow, or difficult to obtain, depending on the shape of the object, the convergence of the far field solution is usually more rapid. In particular, the convergence of the far field is mostly dependent on the size of the object, rather than its shape.