Vandana Dwarka - Resolving Divergence: The First Multigrid Scheme for the Highly Indefinite Helmholtz Equation Using Classical Components.

In this talk, we present the first stand-alone classical multigrid solver for the highly indefinite 2D Helmholtz equation with constant costs per iteration, addressing a longstanding open problem in numerical analysis. Our work covers both large constant and non-constant wavenumbers up to k = 500 in 2D. We obtain a full V − and W−cycle without any level-dependent restrictions. Another powerful feature is that it can be combined with the computationally cheap weighted Jacobi smoother. The key novelty lies in the use of higher-order inter-grid transfer operators. When combined with coarsening on the Complex Shifted Laplacian, rather than the original Helmholtz operator, our solver is h−independent and scales linearly with the wavenumber k. If we use GMRES(3) smoothing we obtain k−independent convergence, and can coarsen on the original Helmholtz operator, as long as the higher-order transfer operators are used.