Analysis of fictitious domain approximations of hard scatterers

• Consider the Helmholtz equation del center dot alpha del p+k(2 alpha)p = 0 in a domain that contains a so-called hard scatterer. The scatterer is represented by the value alpha = epsilon, for 0 < epsilon << 1, whereas alpha = 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 epsilon -> 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 epsilon.

Ämnesord

TEKNIK OCH TEKNOLOGIER  -- Maskinteknik -- Strömningsmekanik och akustik (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Mechanical Engineering -- Fluid Mechanics and Acoustics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Nyckelord

Helmholtz equation

hard scatterers

fictitious domain method

inf-sup condition

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