MATHEMATICAL FOUNDATIONS FOR ERROR ESTIMATION IN NUMERICAL-SOLUTIONS OF INTEGRAL-EQUATIONS IN ELECTROMAGNETICS

The problem of error estimation in the numerical solution of integral equations that arise in electromagnetics is addressed, The direct meth od (Green's theorem or field approach) and the indirect method (layer ansatz or source approach) lead to well-known integral equations both of the first kind [electric field integral equations (EFIE)] and the s econd kind [magnetic field integral equations (MFIE)]. These equations are analyzed systematically in terms of the mapping properties of the integral operators, It Is shown how the assumption that field quantit ies have finite energy leads naturally to describing the mapping prope rties in appropriate Sobolev spaces, These function spaces are demysti fied through simple examples which also are used to demonstrate the im portance of knowing in which space the given data lives and in which s pace the solution should be sought, It is further shown how the method of moments (or Galerkin method) is formulated in these function space s and how residual error can be used to estimate actual error in these spaces, The condition number of all of the impedance matrices that re sult from discretizing the integral equations, including first kind eq uations, is shown to he bounded when the elements are computed appropr iately, Finally, the consequences of carrying out all computations in the space of square integrable functions, a particularly friendly Sobo lev space, are explained.