Multigrid-based preconditioner for heterogeneous, high wavenumber Helmholtz problems¹

Yogi A. Erlangga
Delft Univ. of Technology, Dept. Appl. Math. Anal.,
Mekelweg 8, 2628 CD Delft, The Netherlands
Correspondence: y.a.erlangga@ewi.tudelft.nl

In this paper an iterative method to solve the Helmholtz equation

$$\mathcal{A} u := \left( - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - k^2(x,y) \right) u = f \Omega = \mathbb{R}^2,$$ (1)

with radiation boundary conditions on $\Gamma = \partial \Omega$ is discussed. The iterative method is based on the Krylov subspace methods (in our case Bi-CGSTAB) accelerated by a multigrid method. Since the Helmholtz equation with high wavenumber $k$ arises problems in both smoothing and coarse grid correction, direct application of a standard multigrid method to (1) usually leads to a diverging method. We find that adding an imaginary shift to the Helmholtz equation may result in a good operator that multigrid can handle it efficiently, and that at the same time can also be used as an effective preconditioner for (1).

In its general formulation, the preconditioning operator is defined as

$$\mathcal{M} := - \frac{\partial^2}{\partial x^2} - \frac{\partial... ... y^2} - (\beta_1 - \hat{j} \beta_2) k^2(x,y), \beta_1, \beta_2 \in \mathbb{R}$$ (2)

with $\hat{j} = \sqrt{-1}$., the complex identity.

To investigate the convergence of multigrid as a solver for (2) and as a preconditioner for (1), analyses based on Rigorous Fourier Analysis (RFA) is done. It is found from this analysis that standard multigrid components can still be applied to (2), with a slight modification to the matrix-dependent interpolation operator by de Zeeuw. This interpolation is effective for cases where heterogeneity is present. Furthermore, from RFA we also find that the combination $(\beta_1,\beta_2) = (1.0,0.5)$ results in a robust preconditioner for (1). For this combination, V(1,1) cycle with Jacobi smoother with small underrelaxation factor ( $\omega = 0.5$)can be applied.

Table 1 shows Bi-CGSTAB iterations preconditioned by (2) for Helmholtz problems with constant $k$. CPU time is measured on a Pentium IV Linux PC. These results, especially with $k = 500$, show the robustness of the method.

Table 1: Number of preconditioned Bi-CGSTAB iteration and CPU time (in brackets) to reduce the initial residual by 7 orders

$k$	40	50	100	150	200	500
iter	26	31	52	73	92	250
CPU time	0.21	0.40	3.3	10.8	25.4	425

Problems with heterogeneity, which are of our main interest, will be presented during the talk.