FEM@LLNL | An O(N) Helmholtz Solver by Time-Filtering the Wave Equation

Sponsored by the MFEM project, the FEM@LLNL Seminar Series focuses on finite element research and applications talks of interest to the MFEM community.

On November 4, 2025, Bill Henshaw of RPI presented “An O(N) Helmholtz Solver by Time-Filtering the Wave Equation." An efficient and high-order accurate solver for Helmholtz problems is described. The approach is based on the WaveHoltz algorithm which computes time-harmonic solutions by time-filtering solutions of the wave equation. The wave equation is solved efficiently with implicit time-stepping using as few as five time-steps per period, independent of the mesh size. When multigrid is used to solve the implicit time-stepping equations, the cost of the resulting WaveHoltz scheme scales linearly with the number of grid points N (at fixed frequency) and is thus optimal in CPU-time and memory usage as the mesh is refined. Krylov space solvers such as GMRES are used to accelerate the basic fixed-point iteration. Eigenvector deflation can be used to further improve the convergence. We have implemented the scheme for complex geometry using overset grids with a solver using the Overture framework. Numerical results are given for problems in two and three space dimensions, to second and fourth-order accuracy, and they show the potential of the approach to solve a wide range of large-scale Helmholtz problems.

Learn more about MFEM at https://mfem.org/ and view the seminar speaker lineup at https://mfem.org/seminar/. LLNL-VIDEO-2013579