Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

Abstract

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher-order finite-element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse-grid correction is assumed to be an ideal operator that annihilates the low-frequency error components and leaves the high-frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two-grid convergence factor. While two-grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is "between" two-grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher-order finite element discretizations, including for Q 2 -Q 1 (Taylor-Hood) elements for the Stokes equations. Finally, we present two-grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two-grid and multigrid convergence factors. K E Y W O R D Sfinite-element method, higher-order elements, Jacobi iteration, local Fourier analysis, multigrid

Local Fourier analysis for mixed finite-element methods for the Stokes equations

“…First, while the LFA smoothing factor gives a good prediction of the true convergence factor for the stabilized discretizations with distributive weighted Jacobi and Braess-Sarazin relaxation, it does not for the Uzawa relaxation (in contrast to what is seen for the MAC discretization [18,35]). For no cases does the LFA smoothing factor offer a good prediction of the true convergence behaviour for the (stable) Q 2 − Q 1 discretization, suggesting that the discretization is responsible for the lack of predictivity, consistent with the results in [15,16]. For both stable and stabilized discretizations, we see that standard distributive weighted Jacobi relaxation loses some of its high efficiency, in contrast to what is seen for the MAC scheme [18,35] but that robustness can be restored with an additional relaxation sweep.…”

“…Thus, the symbols of L 1,1 h and L 2,2 h are 4 × 4 matrices. For more details of LFA for the Laplace operator using higher-order finite-element methods, refer to [15]. Similarly to the Laplace operator, both terms in the gradient, (∂ x ) h and (∂ y ) h , can be treated as (4 × 1)-block operators.…”

“…Recently, the validity of LFA has been further analysed [14], extending this exact prediction to a wider class of problems. However, the LFA smoothing factor is also known to lose its predictivity of the true convergence in some cases [15,16,17]. In particular, the smoothing factor of LFA overestimates the two-grid convergence factor for the Taylor-Hood (Q 2 − Q 1 ) discretization of the Stokes equations with Vanka relaxation [16].…”

Convergence analysis for parallel‐in‐time solution of hyperbolic systems

“…The Fourier symbols of M, K, and B are derived from standard calculations 38,39 and given for reference in the Appendix.…”

“…, of the one-dimensional Q2 mass and stiffness matrices, respectively. 38,39 The Fourier symbols of the derivative operators, B x andB , are given by 39 ] , respectively. The Fourier symbols of the time integrators, Φ c and Φ cc , on the first and second coarse grids can be derived analogously, simply by adjusting the value of Δt in the definition ofΦ.…”