Manufactured solutions for the method-of-moments implementation of the electric-field integral equation

Code Verification for Practically Singular Equations

“…However, the integrals in (7) cannot be accurately computed due to the presence of the Green's function, which yields a singularity when x = x . In [1], this challenge is mitigated by manufacturing the Green's function as well. As a result, the matrix becomes practically singular, admitting infinite solutions u h .…”

“…To perform code verification for a singular system of equations, such as that in (12), an approach is presented in [1], which uses a QR factorization of the transpose of the matrix to select the u h closest to u n that satisfies the system of equations. To simplify the factorization, we divide both sides of (12) by h once more:…”

“…The method-of-moments implementation of the electric-field integral equation (EFIE) yields many codeverification challenges due to the various sources of numerical error and their possible interactions [1,2]. Matters are further complicated by singular integrals, which arise from the presence of a Green's function.…”

“…To address these singular integrals, an approach is presented in [1] wherein both the solution and Green's function are manufactured. Because the arising equations are poorly conditioned, they are reformulated as a set of constraints for an optimization problem that selects the solution closest to the manufactured solution.…”

“…In this paper, we demonstrate how, for such practically singular systems of equations, computing the truncation error by inserting the exact solution into the discretized equations cannot detect certain orders of coding errors. On the other hand, the discretization error from the optimal solution [1] is a more sensitive metric.…”