[Paper Review] A closed-form multigrid smoothing factor for an additive Vanka-type smoother applied to the Poisson equation

Chen Greif, Yunhui He|arXiv (Cornell University)|Nov 4, 2021

Advanced Numerical Methods in Computational Mathematics17 references4 citations

TL;DR

This paper derives closed-form expressions for the optimal smoothing factors of additive Vanka-type smoothers applied to the Poisson equation using local Fourier analysis. It reveals that the element-wise Vanka smoother is equivalent to a scaled mass matrix from linear/bilinear finite element methods, leading to a highly efficient mass-based relaxation scheme that achieves low smoothing factors with only one matrix-vector product, offering a fast, sparse, and well-conditioned alternative to traditional Vanka smoothers.

ABSTRACT

We consider an additive Vanka-type smoother for the Poisson equation discretized by the standard finite difference centered scheme. Using local Fourier analysis, we derive analytical formulas for the optimal smoothing factors for two types of smoothers, called vertex-wise and element-wise Vanka smoothers, and present the corresponding stencils. Interestingly, in one dimension the element-wise Vanka smoother is equivalent to the scaled mass operator obtained from the linear finite element method, and in two dimensions the element-wise Vanka smoother is equivalent to the scaled mass operator discretized by bilinear finite element method plus a scaled identity operator. Based on these discoveries, the mass matrix obtained from finite element method can be used as an approximation to the inverse of the Laplacian, and the resulting mass-based relaxation scheme features small smoothing factors in one, two, and three dimensions. Advantages of the mass operator are that the operator is sparse and well conditioned, and the computational cost of the relaxation scheme is only one matrix-vector product; there is no need to compute the inverse of a matrix. These findings may help better understand the efficiency of additive Vanka smoothers and develop fast solvers for numerical solutions of partial differential equations.

Motivation & Objective

• To theoretically analyze the convergence behavior of additive Vanka smoothers for the Poisson equation using local Fourier analysis.
• To derive closed-form expressions for the optimal smoothing factors of vertex-wise and element-wise Vanka smoothers in 1D and 2D.
• To identify the mathematical equivalence between the element-wise Vanka smoother and scaled finite element mass matrices.
• To propose a mass-based relaxation scheme as a computationally efficient alternative to standard additive Vanka smoothers.
• To validate the analytical results numerically and demonstrate the effectiveness of the proposed scheme in multigrid solvers.

Proposed method

• Applies local Fourier analysis (LFA) to compute the smoothing factor for additive Vanka-type smoothers on structured grids.
• Derives analytical formulas for optimal relaxation parameters and smoothing factors in 1D and 2D for both vertex-wise and element-wise Vanka patches.
• Identifies that the element-wise Vanka smoother in 1D is equivalent to the scaled mass matrix from linear finite elements, and in 2D to the bilinear finite element mass matrix plus a scaled identity.
• Proposes a mass-based relaxation scheme using the finite element mass matrix as an approximation to the inverse Laplacian, avoiding subproblem solves.
• Validates the analytical predictions via numerical experiments, computing LFA two-grid convergence factors and comparing them with smoothing factors.
• Extends the analysis to 3D by proposing a mass-based smoother using the trilinear finite element mass matrix.

Figure 2: Eigenvalue distribution of $\widetilde{E}$ for Vanka-type method with $\nu=1$ . Left: element-wise Vanka. Right: vertex-wise Vanka.

Experimental results

Research questions

• RQ1 What are the closed-form expressions for the optimal smoothing factors of additive Vanka smoothers applied to the Poisson equation in 1D and 2D?
• RQ2 How do the stencils of vertex-wise and element-wise Vanka smoothers relate to finite element mass matrices?
• RQ3 Can the finite element mass matrix be used as an effective and efficient alternative to the standard additive Vanka smoother?
• RQ4 Why does the element-wise Vanka smoother achieve such low smoothing factors, and what is its theoretical basis?
• RQ5 How does the proposed mass-based relaxation scheme compare in performance to traditional Vanka smoothers in terms of convergence and computational cost?

Key findings

• The optimal smoothing factor for the element-wise Vanka smoother in 1D is 12/17 ≈ 0.706, and in 2D it is 24/25 = 0.96, indicating excellent smoothing performance.
• In 1D, the element-wise Vanka smoother is mathematically equivalent to the scaled mass matrix from linear finite elements.
• In 2D, the element-wise Vanka smoother is equivalent to the bilinear finite element mass matrix plus a scaled identity operator.
• The proposed mass-based relaxation scheme achieves a two-grid convergence factor of 0.333 in 2D with ω = 3/4, matching the theoretical smoothing factor.
• The vertex-wise Vanka smoother in 1D has a higher smoothing factor of 81/104 ≈ 0.779, but the LFA two-grid convergence factor is 0.067, indicating a small gap between smoothing factor and actual convergence.
• The mass-based smoother in 3D achieves a two-grid convergence factor of 0.618 with ω = 729/848, showing strong performance despite the increased dimensionality.

Figure 3: Element-wise Vanka with $\nu=1$ . Left: the magnitude of eigenvalues of $\widetilde{E}_{e}$ as a function of the Fourier modes, $(\theta_{1},\theta_{2})$ . Right: the magnitude of eigenvalues of $\widetilde{S}_{e}$ as a function of the Fourier modes, $(\theta_{1},\theta_{2})$ .