Electrostatic Capacity of a Metallic Cylinder: Effect of the Moment Method Discretization Process on the Performances of the Krylov Subspace Techniques

When a straight cylindrical conductor of finite length is electrostatically charged, its electrostatic potential ϕ depends on the electrostatic charge q e , as expressed by the equation L ( q e ) = ϕ , where L is an integral operator. Method of moments (MoM) is an excellent candidate for solving L ( q e ) = ϕ numerically. In fact, considering q e as a piece-wise constant over the length of the conductor, it can be expressed as a finite series of weighted basis functions, q e = ∑ n = 1 N α n f n (with weights α n and N , number of the subsections of the conductor) defined in the L domain so that ϕ becomes a finite sum of integrals from which, considering testing functions suitably combined with the basis functions, one obtains an algebraic system L m n α n = g m with dense matrix, equivalent to L ( q e ) = ϕ . Once solved, the linear algebraic system gets α n and therefore q e is obtainable so that the electrostatic capacitance C = q e / V , where V is the external electrical tension applied, can give the corresponding electrostatic capacitance. In this paper, a comparison was made among some Krylov subspace method-based procedures to solve L m n α n = g m . These methods have, as a basic idea, the projection of a problem related to a matrix A ∈ R n × n , having a number of non-null elements of the order of n , in a subspace of lower order. This reduces the computational complexity of the algorithms for solving linear algebraic systems in which the matrix is dense. Five cases were identified to determine L m n according to the type of basis-testing functions pair used. In particular: (1) pulse function as the basis function and delta function as the testing function; (2) pulse function as the basis function as well as testing function; (3) triangular function as the basis function and delta function as the testing function; (4) triangular function as the basis function and pulse function as the testing function; (5) triangular function as the basis function with the Galerkin Procedure. Therefore, five L m n and five pair q e and C were computed. For each case, for the resolution of L m n α n = g m obtained, GMRES, CGS, and BicGStab algorithms (based on Krylov subspaces approach) were implemented in the MatLab® Toolbox to evaluate q e and C as N increases, highlighting asymptotical behaviors of the procedures. Then, a particular value for N is obtained, exploiting both the conditioning number of L m n and considerations on C , to avoid instability phenomena. The performances of the exploited procedures have been evaluated in terms of convergence speed and CPU-times as the length/diameter and N increase. The results show the superiority of BcGStab, compared to the other procedures used, since even if the number of iterations increases significantly, the CPU-time decreases (more than 50%) when the asymptotic behavior of all the procedures is in place. This superiority is much more evident when the CPU-time of BicGStab is compared with that achieved by exploiting Gauss elimination and Gauss–Seidel approaches.