A controllability method with an artificial boundary condition for the exterior Helmholtz problem
The author presents a numerical resolution of the outer Helmholtz problem : \( -\Delta u+k^{2}u=F\) in an unbounded domain \(\Omega \) of \({\mathbb R}^{d}\) (\( d=2,3\)), with the homogeneous Dirichlet boundary conditions: \(U=0\) on \( \partial \Omega \) and the\ Sommerfeld radiation condition at infinity : \( \lim_{r\rightarrow +\infty }r^{(d-1)/2}\left( \partial U/\partial r+kU\right) =0\). \(F\in L^{2}\left( \Omega \right) \) is supposed to have a compact support. The author uses a controllability approach for a wave equation posed in the bounded domain \(\Omega _{a}\times \left( 0,T\right) \), with \(\Omega _{a}=\left\{ x\in {\mathbb R}^{d}\mid \left| x\right| <a\right\} \) (\(a>0\)), as previously introduced by \textit{M. O. Bristeau} et al [J. Comput. Phys. 147, No. 2, 265--292 (1998; Zbl 0926.65054)]. This approach requires the introduction of an extra and artificial boundary condition : \(\partial u/\partial n+\partial u/\partial t=-{\mathcal S}u-iku\) on \(\Gamma _{a}\times \left( 0,T\right) \), with \(\Gamma _{a}=\left\{ x\in {\mathbb R}^{d}\mid \left| x\right| =a\right\} \), where \({\mathcal S}\) is a Dirichlet-to-Neumann operator. \({\mathcal S}\varphi \)\ is indeed the flux through \(\Gamma _{a}\) of the solution of an outer Helmholtz problem similar to the original one, but posed in \(\Omega _{a}^{\prime }=\left\{ x\in {\mathbb R }^{d}\mid \left| x\right| >a\right\} \) and with Dirichlet boundary condition \(\varphi \) on \(\Gamma _{a}\). The author first proves the existence of a solution in \(H_{loc}^{2}\left( \Omega \right) \) of the original Helmholtz problem. He then proves the equivalence of this problem with a controllability problem. For the numerical resolution of this wave equation, he introduces a semi-discrete algorithm. The proof of the uniqueness of the wave equation requires a condition on the subspaces \(V_{h}\). The last part of the paper presents some numerical examples : scattering by a disk or by a \(\Pi \)-shaped resonator.
scientific article; zbMATH DE number 1933153
scientific article