Fictitious domains methods for wave diffraction

Let us consider as a model problem the propagation of waves in an exterior domain, defined as the complement of a bounded object. Solving this problem with a finite difference method requires a staircase approximation of the boundary of the object (see Figure 12.1). The great disadvantage of this approach is that spurious numerical diffractions are introduced in the solution from the inadequate geometric approximation. To avoid this drawback one can use a finite element method with a mesh that follows exactly the boundary of the object (see Figure 12.1). To have a method which is comparable in cost with the finite defferences method one should use finite elements with mass-lumping (see section 10.4 and cf. [48]).