A robust domain decomposition method for the Helmholtz equation with high wave number (Q2814664)

The paper starts by the Helmholtz equation in \(\mathbb{R}^{2}\) divided in two half-planes. The iterative method consists of solving the Helmholtz equation in alternating half-planes, with the solution transmitted on the common boundary by Robin boundary conditions with different coefficients from the two sides. In addition, the iteration are underrelaxed. By optimizing the coefficiencts, a convergence bound is obtained which is independent of the wave number \(k\) for \(kh=\mathrm{constant}\), and of the mesh size \(h\) for fixed \(k\). The convergence rate is estimated by Fourier analysis. The authors then describe implementation on a bounded domain and confirm the convergence rates numerically. Extension of the method to multiple subdomains is also described, which shows independence on \(h\) for constant \(k\), but the number of iterations grows with the number of subdomains. There is no coarse problem.