  The Helmholtz equation is related to seismic exploration, sonar, antennas,
and medical imaging applications. It is one of the most challenging problems to
solve in terms of accuracy and convergence due to the scalability issues of the
numerical solvers. For 3D large-scale applications, high-performance parallel
solvers are also needed. In this paper, a matrix-free parallel iterative solver
is presented for the three-dimensional (3D) heterogeneous Helmholtz equation.
We consider the preconditioned Krylov subspace methods for solving the linear
system obtained from finite-difference discretization. The Complex Shifted
Laplace Preconditioner (CSLP) is employed since it results in a linear increase
in the number of iterations as a function of the wavenumber. The preconditioner
is approximately inverted using one parallel 3D multigrid cycle. For parallel
computing, the global domain is partitioned blockwise. The matrix-vector
multiplication and preconditioning operator are implemented in a matrix-free
way instead of constructing large, memory-consuming coefficient matrices.
Numerical experiments of 3D model problems demonstrate the robustness and
outstanding strong scaling of our matrix-free parallel solution method.
Moreover, the weak parallel scalability indicates our approach is suitable for
realistic 3D heterogeneous Helmholtz problems with minimized pollution error.

Numerical Analysis;Numerical Analysis