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

Example

Use the four-interval mesh from the homogeneous weak Dirichlet Poisson form on $(0, 1)$ and the exact piecewise-linear coarse correction. Here $A_{h} = 4 T_{3}$ , where $T_{3} = (\begin{array}{ccc} 2 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array})$ , and $P = (1 / 2, 1, 1 / 2)^{T}$ . For the error $e = (1, 0, - 1)^{T}$ , the coarse correction leaves the vector unchanged, while one Richardson sweep with step $1 / λ_{\max} (A_{h})$ damps it by $\sqrt{2} - 1$ . The symmetric two-grid error map therefore sends this particular mode to $(3 - 2 \sqrt{2}) e$ . The level-uniform theorem bounds the full operator norm by $2 / 3$ for one sweep, not merely this single mode.

The calculation uses the homogeneous weak-form convention and makes no claim about solution existence or finite-element discretization error.

Facts & Assumptions

Given: The homogeneous weak Dirichlet Poisson form on $(0, 1)$ , the four-interval fine mesh with $A_{h} = 4 T_{3}$ , $T_{3} = (\begin{array}{ccc} 2 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array})$ , coarse prolongation $P = (1 / 2, 1, 1 / 2)^{T}$ , the error $e = (1, 0, - 1)^{T}$ , and one Richardson sweep.

[F1]

Weak Dirichlet solutions for a divergence-form operator: the homogeneous weak Dirichlet problem uses $u \in H_{0}^{1} (0, 1)$ and the form identity against every $H_{0}^{1}$ test function; this branch assumes Countable Choice.

[F2]

Uniformly elliptic divergence-form operators and their sesquilinear forms: for the real Poisson coefficients $a^{11} = 1$ , $b = c = 0$ , the form is $a (u, v) = \int_{0}^{1} u^{'} v^{'} d x$ .

[F3]

Galerkin coarse matrices and the exact coarse-correction error map: $A_{H} = P^{T} A_{h} P$ and $C = I - P A_{H}^{- 1} P^{T} A_{h}$ when $A_{H}$ is invertible.

[F4]

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

[F5]

The symmetric 1D Poisson two-grid cycle is uniformly contractive: the symmetric 1D Poisson map has $∥ E ∥_{A_{h}} \leq 2 / (2 ν + 1)$ for every $ν \geq 1$ and every even $N \geq 4$ .

[F6]

Continuous piecewise-affine functions with zero endpoints lie in H1-zero: each continuous piecewise-affine function on $[0, 1]$ vanishing at $0$ and $1$ belongs to $H_{0}^{1} (0, 1)$ .

[A1]

Assume The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the homogeneous Sobolev interface in [F1]. This example uses the homogeneous branch and makes no inhomogeneous trace claim.

Verification

1.1givenA1F1F2F6algebra

The fine nodal hats are continuous piecewise affine and vanish at both endpoints, so [F6] puts them in the homogeneous test space of [F1] under [A1]; [F2] then gives their form as $\int_{0}^{1} u^{'} v^{'}$ . Direct determinant expansion gives $\det (t I - T_{3}) = t^{3} - 6 t^{2} + 10 t - 4 = (t - 2) (t^{2} - 4 t + 2),$ so $λ_{\max} (A_{h}) = 4 (2 + \sqrt{2})$ , while direct multiplication gives $A_{h} e = 8 e$ . Thus one pre-sweep $S = I - A_{h} / λ_{\max} (A_{h})$ sends $e$ to $(1 - 8 / [4 (2 + \sqrt{2})]) e = (\sqrt{2} - 1) e$ .

2.1F3F4step 1.1algebra

Direct multiplication gives $A_{h} P = (0, 4, 0)^{T}$ and $A_{H} = P^{T} A_{h} P = 4$ . Hence $C = I - P (0, 1, 0)$ and $C e = e$ , since the middle component of $e$ is zero. Since $S = I - A_{h} / λ_{\max}$ is symmetric and commutes with $A_{h}$ , its $A_{h}$ -adjoint $A_{h}^{- 1} S^{T} A_{h}$ equals $S$ . Thus [F4] gives $E e = (\sqrt{2} - 1)^{2} e = (3 - 2 \sqrt{2}) e$ .

3.1F4F5step 2.1algebra∎

Since $3 - 2 \sqrt{2} < 2 / 3$ (equivalently $49 < 72$ after multiplying by positive quantities and squaring), this mode is damped by less than $2 / 3$ . [F5] with $ν = 1$ bounds the full energy operator norm by $2 / 3$ , including every fine-grid mode.

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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)