Wave phenomena play an important role in many different applications such as MRI scans, seismology and acoustics [41, 49, 47]. At the core of such applications lies the Helmholtz equation, which represents the time-independent version of the wave equation. Simulating a Helmholtz problem numerically with accurate numerical solutions for large wave numbers is challenging. Numerical solvers for the Helmholtz problem have to balance having accurate numerical solutions, requiring a number of iterations to reach convergence that is independent of the wave number and solving with linear time complexity with respect to the grid nodes. Currently, there is no numerical Helmholtz solver that can satisfy these requirements at once.
We developed Schwarz domain decomposition preconditioners which leads to wave number independent convergence for wave numbers in 2D and 3D, while remaining to have accurate numerical solutions. The preconditioners use two-level Schwarz preconditioners, with the coarse problem being constructed using higher-order interpolation with quadratic rational Bézier curves. The developed domain decomposition preconditioners are designed to leverage parallel computing in the future in an attempt for the preconditioners to acquire the ability to solve with linear time complexity.

In this research, the preconditioner resulting in wave number independent convergence and the lowest iteration count is the two-level scaled hybrid Schwarz preconditioner with a coarse problem constructed using higher-order Bézier interpolation. This preconditioner uses a deflation method to remove unwanted eigenvalues. Removing these unwanted eigenvalues results in a clustering of the eigenvalues which is more favourable for GMRES. Currently, all the developed preconditioners suffer from high computational cost for large wave numbers, due to the coarse problem becoming large. Decreasing the coarse problem size of the preconditioners, while remaining to have wave number independent convergence, has shown to been unsuccessful. To better understand the required conditions for wave number independent convergence of the preconditioners, we investigated the relationship between the number of coarse grid nodes and the wave length, too see if there is anything generalizable about this relationship and wave number independent convergence of the solvers.
In conclusion, the balancing for a Helmholtz solver to have accurate numerical solutions, requiring a number of iterations to reach convergence that is independent of the wave number and solving with linear time complexity
is again shown to be difficult. This work provides the initial development and testing of promising wave number independent Helmholtz solvers, from which more research should follow that tackle its biggest computational problems.