2015/01/01 by Fotios Kasolis, Eddie Wadbro, Martin Berggren · 2 citations
Engineering · Computer Science · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics
paper · doi:10.1137/140981630
Consider the Helmholtz equation ∇⋅α∇ p + k2α p = 0 in a domain that contains a so-called hard scatterer. The scatterer is represented by the value α = ε, for 0<ε≪ 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 ε→ 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 ε.