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The symmetric 1D Poisson two-grid cycle is uniformly contractive

Statement

For the nested one-dimensional Poisson grids and prolongation of Level-uniform 1D Poisson smoothing and approximation bounds, take the pre-smoothing error map to be $S^{ν}$ , with $S = I - A_{h} / λ_{\max} (A_{h})$ , and take the post-smoother to be its $A_{h}$ -adjoint. The resulting symmetric two-grid error map satisfies

$∥ E ∥_{A_{h}} \leq \frac{2}{2 ν + 1} \leq \frac{2}{3} < 1 (ν \geq 1),$

uniformly for all even $N = 2 M \geq 4$ . This proves one concrete level-uniform two-grid rate; it does not assert convergence of an arbitrary AMG hierarchy.

Facts & Assumptions

Given: The one-dimensional Poisson matrices, piecewise-linear prolongation, exact Galerkin correction, and $ν \geq 1$ Richardson sweeps specified in Level-uniform 1D Poisson smoothing and approximation bounds.

[F1]

Level-uniform 1D Poisson smoothing and approximation bounds: $∥ C S^{ν} e ∥_{A_{h}}^{2} \leq 2 (2 ν + 1)^{- 1} ∥ e ∥_{A_{h}}^{2}$ for every fine error $e$ .

[F2]

The symmetric two-grid error map from a pre-smoother and Galerkin correction: the symmetric two-grid map is $E = (S^{ν})^{*_{A_{h}}} C S^{ν}$ .

[F3]

A joint smoothing-correction estimate gives symmetric two-grid energy contraction: a joint estimate with $0 \leq ρ < 1$ implies $∥ E ∥_{A_{h}} \leq ρ$ .

Proof

1.1givenF1algebra

By [F1], the pre-smoother $S^{ν}$ satisfies the joint estimate with $ρ = 2 / (2 ν + 1)$ . For $ν \geq 1$ , $0 < ρ \leq 2 / 3 < 1$ .

2.1F2F3step 1.1∎

Apply [F3] to the symmetric error map [F2] with this value of $ρ$ . It follows that $∥ E ∥_{A_{h}} \leq 2 / (2 ν + 1) \leq 2 / 3$ , independently of $N$ .

Depends on

Used by

One high-frequency mode in the finite one-dimensional Poisson two-grid cycle Example

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Chapter 6 (standard reference, not scraped)
Numerical Methods II, University of Heidelberg lecture notes, Chapter 10 (standard reference, not scraped)