Analysis of Fictitious Domain Approximations of Hard Scatterers

Consider the Helmholtz equation ∇ · α∇p+ k2αp = 0 in a domain that contains a so-called hard scatterer. The scatterer is represented by the value α = , for 0 < 1, whereas α = 1 whenever the scatterer is absent. This scatterer model is often used for the purpose of design optimization and constitutes a fictitious domain approximation of a body characterized by homogeneous Neumann conditions on its boundary. However, such an approximation results in spurious resonances inside the scatterer at certain frequencies and causes, after discretization, ill-conditioned system matrices. Here, we present a stabilization strategy that removes these resonances. Furthermore, we prove that, in the limit → 0, the stabilized problem provides linearly convergent approximations of the solution to the problem with an exactly modeled scatterer. Numerical experiments indicate that a finite element approximation of the stabilized problem is free from internal resonances, and they also suggest that the convergence rate is indeed linear with respect to .