Convergence of Waveholtz for variable wave speeds

Establish whether the Waveholtz iteration converges for the Helmholtz equation with variable wave speed, i.e., for the variable-coefficient operator ∇·(c(x)^2∇u) + ω^2 u = f, and identify precise conditions on c(x) and function spaces under which convergence holds.

Background

The paper provides a convergence proof for the Waveholtz method in the constant-coefficient case on R^d using weighted Sobolev spaces and limiting-absorption techniques. Many practical applications involve spatially varying wave speeds c(x), for which the present analysis does not apply.

The authors explicitly note that extending the convergence proof to variable wave speeds remains unproven, highlighting a key theoretical gap between numerical evidence and existing analysis.