Theoretical analysis demonstrates a $1/\sqrt{n}$ error decay for WaveHoltz iterates in $d$-dimensional space, proving convergence to outgoing Helmholtz solutions.
Key Points
Analyze the convergence behavior of the WaveHoltz iteration for solving the constant-coefficient Helmholtz equation across whole Euclidean space.
Characterized the residual error between WaveHoltz iterates and outgoing Helmholtz solutions as a driven Helmholtz equation with a specific forcing term.
Derived frequency-explicit mathematical bounds in weighted Sobolev norms to quantify error reduction as a function of iteration count.
Proved that the difference between the iterate and the outgoing solution decreases at an asymptotic rate of $1/\sqrt{n}$ with iteration number $n$.
Confirmed the theoretical convergence of the real part of the WaveHoltz iterates to the real part of the outgoing Helmholtz solution.