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Grid and common-space stability agree under uniform norm comparison

Statement

Let a finite-grid lift be as in Finite-grid restriction, reconstruction, and common-space lift over the common Banach-space interface of Common Banach-space interface for a linear IVP and its time schemes. Assume there are constants $a, b, P > 0$ , independent of $ν$ , such that for every $g \in G_{ν}$ and $x \in X$ , $a ∥ g ∥_{G_{ν}} \leq ∥ R_{ν} g ∥_{X} \leq b ∥ g ∥_{G_{ν}}, ∥ P_{ν} ∥_{X \to G_{ν}} \leq P .$ Then the lifted family $(S_{ν})$ is stable on finite time intervals if and only if the grid-update family $(Q_{ν})$ is stable in the declared grid norms on finite time intervals.

Here grid stability means that for every $T > 0$ there is $C_{T} < \infty$ such that $∥ Q_{ν}^{n} ∥_{G_{ν} \to G_{ν}} \leq C_{T}$ whenever $ν, n \in N$ and $n τ_{ν} \leq T$ .

Facts & Assumptions

Given: The common-space and grid-lift data and uniform constants $a, b, P$ from the statement.

[F1]

The grid lift has $P_{ν} R_{ν} = I_{G_{ν}}$ , $U (τ_{ν}) (\ker P_{ν}) \subseteq \ker P_{ν}$ , and $S_{ν} = R_{ν} Q_{ν} P_{ν} + U (τ_{ν}) (I_{X} - R_{ν} P_{ν})$ (Finite-grid restriction, reconstruction, and common-space lift).

[F2]

The exact semigroup satisfies $∥ U (t) ∥_{o p} \leq M_{T}$ for $0 \leq t \leq T$ (Common Banach-space interface for a linear IVP and its time schemes).

[F3]

The operator norm is the supremum over the unit ball, so $∥ T x ∥ \leq ∥ T ∥_{o p} ∥ x ∥$ for every $x$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[F4]

A property true at $0$ and preserved by successor holds for every $n \in N$ (The principle of mathematical induction).

Proof

1.1F1givenalgebra

The identity $P_{ν} R_{ν} = I$ gives $x = R_{ν} P_{ν} x + (I_{X} - R_{ν} P_{ν}) x .$ The first term is in $R_{ν} (G_{ν})$ and the second is in $\ker P_{ν}$ . For $g \in G_{ν}$ , the lift satisfies $S_{ν} R_{ν} g = R_{ν} Q_{ν} g$ ; for $y \in \ker P_{ν}$ , it satisfies $S_{ν} y = U (τ_{ν}) y$ , which remains in $\ker P_{ν}$ by [F1]. Thus both summands are invariant under $S_{ν}$ .

2.1F1F4step 1.1algebra

Induction [F4] on $n \in N$ , using the two invariant summands from step 1.1 and the semigroup identity $U (τ_{ν})^{n} = U (n τ_{ν})$ , gives $S_{ν}^{n} = R_{ν} Q_{ν}^{n} P_{ν} + U (n τ_{ν}) (I_{X} - R_{ν} P_{ν}) .$ For $n = 0$ the right side is $R_{ν} P_{ν} + (I_{X} - R_{ν} P_{ν}) = I_{X}$ ; the induction step applies $R_{ν} Q_{ν}$ on the first summand and $U (τ_{ν})$ on the second.

3.1F1F2F3step 2.1givenalgebra

Suppose the grid updates are stable, with $∥ Q_{ν}^{n} ∥_{G_{ν} \to G_{ν}} \leq C_{T}$ whenever $n τ_{ν} \leq T$ . For every $x \in X$ , step 2.1 and the comparison bounds give $∥ S_{ν}^{n} x ∥_{X} \leq b C_{T} P ∥ x ∥_{X} + M_{T} (1 + b P) ∥ x ∥_{X}$ : indeed, $∥ R_{ν} Q_{ν}^{n} P_{ν} x ∥_{X} \leq b C_{T} P ∥ x ∥_{X}$ and $∥ U (n τ_{ν}) (I_{X} - R_{ν} P_{ν}) x ∥_{X} \leq M_{T} (1 + b P) ∥ x ∥_{X}$ . Taking the supremum over $∥ x ∥_{X} \leq 1$ using [F3] proves the uniform common-space power bound, hence stability.

4.1F3step 1.1givenalgebra∎

Conversely, suppose $∥ S_{ν}^{n} ∥_{X \to X} \leq C_{T}$ for $n τ_{ν} \leq T$ . For every $g \in G_{ν}$ , step 1.1 gives $S_{ν}^{n} R_{ν} g = R_{ν} Q_{ν}^{n} g$ . The lower and upper norm comparisons and [F3] therefore give $a ∥ Q_{ν}^{n} g ∥_{G_{ν}} \leq ∥ R_{ν} Q_{ν}^{n} g ∥_{X} = ∥ S_{ν}^{n} R_{ν} g ∥_{X} \leq C_{T} ∥ R_{ν} g ∥_{X} \leq C_{T} b ∥ g ∥_{G_{ν}} .$ Thus $∥ Q_{ν}^{n} ∥_{G_{ν} \to G_{ν}} \leq C_{T} b / a$ uniformly on the same finite interval. The two stability notions are equivalent under the stated uniform comparisons.

Depends on

Used by

A mesh-dependent grid norm hides common-space instability Example

Dependency tree · two levels

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Sources

Lloyd N. Trefethen, Finite Difference and Spectral Methods, Chapter 4, §4.2 (standard reference, not scraped)
R. J. LeVeque, AMath 585–6 Notes, Chapter 12, §§12.1–12.5.1 (standard reference, not scraped)