A fast spectral element solver combining static condensation and multigrid techniques

We propose a spectral element multigrid method for the two-dimensional Helmholtz equation discretized on regular grids. Combining p-multigrid with static condensation the method achieves nearly linear complexity with an order-independent convergence rate for solving the condensed equations. For smoothing we consider two groups of edge-based relaxation schemes, the best of which attains a multigrid convergence rate of ρ ≈ 0.014 to 0.028. Numerical experiments have been carried out that demonstrate the robustness of the approach for orders up to 32 and a total of 109 degrees of freedom. In comparison with a fast finite difference solver, the latter is clearly outperformed already for errors of one percent or lower.