Abstract

Multigrid methods play a key role in large-scale scientific simulation because they are among the fastest and most scalable approaches for solving systems of equations. They are widely used to solve the sparse linear systems that arise in these simulations, and they have been shown to scale efficiently on today’s supercomputers. However, this success has been primarily for symmetric positive definite systems. Developing effective methods for indefinite systems such as the Helmholtz equation remains an open problem. Several issues must be addressed: the system has both positive and negative eigenvalues (or complex), requiring special treatment in the smoother; the near-kernel is oscillatory, breaking the standard geometric smoothness assumption; and the coarse-grid correction does not form a projection. In this talk, we present recent research aimed at addressing these issues. We also provide numerical results for both Helmholtz and a shifted Laplacian problem with extremely large shifts.

Video Recording