Boundary Element Solution of Scattering Problems Relative to a Generalized Impedance Boundary Condition
This paper addresses the solution of a scattering problem governed by the Helmholtz equation and relative to a boundary condition expressed through a second-order differential operator by a boundary element method. The usual approach, setting a boundary integral equation from a representation of the solution, cannot be directly applied then since derivatives of single- or double-layer potentials are not given explicitly by integrals converging in the usual sense. As a result, it is not suited for effective numerical computations. First, a suitable procedure bypasses the difficulty at the cost of tripling the number of unknowns relative to the case of a zero impedance, that is, with a Neumann boundary condition. In fact, a suitable lumping process in the computation of an integral permits to eliminate the two supplementary unknowns in the assembly process at the element level. Furthermore, the final linear system to be solved has a symmetric matrix exactly as the one involved in the double-layer solution of the problem relative to a Neumann boundary condition. Moreover, the former appears as a small perturbation of the latter for a nearly vanishing impedance.