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Fictitious domain methods for the numerical solution of two-dimensional scattering problems. (English) Zbl 0909.65119
The authors consider the two-dimensional exterior Helmholtz equation $\begin{gathered} \triangle u+\omega^2u=0\;\text{in} \mathbb{R}^2\backslash\overline\Omega,\quad u=-v_I\;\text{on} \partial\Omega\\ \lim_{r\to\infty}\sqrt{r}\left(\frac{\partial u}{\partial r}-{\mathbf i}\omega u\right)=0, \end{gathered}$ which can be used to model the scattering of time-harmonic electromagnetic or acoustic waves by an obstacle denoted by $$\Omega$$. Here, $$v_I({\mathbf x})=e^{i\mathbf{\omega}\cdot{\mathbf x}}$$ is the time-harmonic incident plane wave the direction of propagation of which is given by the vector $${\mathbf \omega}$$. This exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. First-order, second-order, and exact nonreflecting boundary conditions are tested on rectangular and circular boundaries. The finite element discretizations of the corresponding approximate boundary value problems are performed using locally fitted meshes, and the discrete equations are solved with fictious domain methods. A special finite element method uses the macro-hybrid formulation based on domain decomposition to couple polar and cartesian coordinate systems. A special preconditioner based on fictitious domains is introduced for the arising algebraic saddle-point system such that the subspace of constraints becomes invariant with respect to the preconditioned iterative procedure. The performance of the new method is compared to the fictitious domain methods both with respect to accuracy and computational cost.
Reviewer: K.Najzar (Praha)

##### MSC:
 65Z05 Applications to the sciences 78A45 Diffraction, scattering 76Q05 Hydro- and aero-acoustics 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 35Q60 PDEs in connection with optics and electromagnetic theory 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35Q35 PDEs in connection with fluid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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