Convergence of the Semi-Discrete WaveHoltz Iteration

TL;DR

The paper analyzes the convergence of the WaveHoltz iteration applied to semi-discretizations of the wave equation aimed at solving the Helmholtz problem. It proves that, under stability and non-resonance (no eigenvalues at ±iω), the iteration converges to the Helmholtz solution with a rate determined by the parabolic distance of discretization eigenvalues from ±iω, and provides iteration-count estimates. The methodology connects time-domain wave dynamics to the frequency-domain problem and recasts WaveHoltz as a left preconditioner for Helmholtz, with concrete results demonstrated for finite-difference and discontinuous Galerkin discretizations in 1D and 2D, including scenarios with impedance boundaries and GMRES acceleration. The findings support robust, scalable Helmholtz solves that leverage efficient wave-equation solvers, with practical iteration counts that scale predictably with frequency and boundary conditions. Potential extensions include time-discrete analyses and Krylov-accelerated schemes.

Abstract

In this paper we prove that for stable semi-discretizations of the wave equation for the WaveHoltz iteration is guaranteed to converge to an approximate solution of the corresponding frequency domain problem, if it exists. We show that for certain classes of frequency domain problems, the WaveHoltz iteration without acceleration converges in $O(ω)$ iterations with the constant factor depending logarithmically on the desired tolerance. We conjecture that the Helmholtz problem in open domains with no trapping waves is one such class of problems and we provide numerical examples in one and two dimensions using finite differences and discontinuous Galerkin discretizations which demonstrate these converge results.

Figures (9)

• Figure 1: The three figures show the modulus of $\hat{\beta}$ in the left half-plane (left), its level contours near $z=i$ which are approximately parabolic (middle), and its restriction to the imaginary axis (right)
• Figure 2: The regions $D$, $G_{d_0}$, and $G'_{d_0}$, for some $d_0 < \alpha$.
• Figure 3: The eigenvalues of the discreization matrix $A$ scaled by $1/\omega$ for finte differences (left) and DG with $P=1$ (middle) and $P=2$ (right). Because $A$ is real, all of its eigenvalues are either real or appear in complex conjugate pairs, so we plot the eigenvalues where $\Im\{\lambda\} \geq 0$. The diamonds highlight the eigenvalues minimizing the parabolic distance of which we draw three level sets. As $\omega$ increases we observe the distance between $i$ and the nearest scaled eigenvalue decreasing. Note that the imaginary part is much larger in magnitude than the real part.
• Figure 4: We display $\varepsilon$ vs. $\omega$ (left), and the condition number $\kappa(R)$ vs. $\omega$ (right). The slopes of the dashed lines for both plots were estimated by regression.
• Figure 5: The left figure shows the relative error (solid), the eigenvector coefficient error (dashed), and the GMRES-accelerated error (dotted), with blue, orange, and green representing $\omega=10\pi, 20\pi, 30\pi$. The right figure reports the number of iterations required for each method to achieve $\|\hat{e}_h^{(n)}\| \leq 10^{-8}\|\hat{e}_h^{(0)}\|$.

Theorems & Definitions (6)

• Theorem 1
• Corollary 1
• proof
• Lemma 1
• Lemma 2
• proof