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A mean-zero square pressure time test

Example

Assume Countable Choice. Take $D = (0, 1)^{2}$ , $T = 1$ , and a smooth MINI observation interface with three time cells $0 = t_{0} < t_{1} < t_{2} < t_{3} = 1$ satisfying the accumulated-pressure proposition's strong-solution and time-step hypotheses. Use the mean-zero pressure sensor $ξ (x, y) = x - \frac{1}{2}$ . Choose any nonconstant $\hat{η} \in C_{c}^{\infty} ((\frac{1}{4}, \frac{3}{4}))$ and set $η = \hat{η} / ∥ {\hat{η}}^{'} ∥_{L^{1} (0, 1)}$ . Then $∥ ξ ∥_{L^{2} (D)} = 1 / \sqrt{12}$ and $∥ η^{'} ∥_{L^{1} (0, 1)} = 1$ . For every node $m \in {0, 1, 2, 3}$ , the accumulated pressure observation error is at most $K H / \sqrt{12}$ and the time-tested pressure observation error is at most $(K + 1) H / \sqrt{12}$ , where $K$ and $H$ are exactly those in the accumulated-pressure proposition.

The diagram records only the definitions: the horizontal cells carry the right-constant pressure reconstruction, and each arrow adds the corresponding pressure increment to the Q-valued primitive. Horizontal spacing is schematic and does not encode the time-step sizes; node heights are layout only and do not encode pressure magnitudes or data.

\begin{tikzpicture}[>=stealth] \node (p0) at (0,1.25) { $P_{h}^{0} = 0$ }; \node (p1) at (2,1.25) { $P_{h}^{1}$ }; \node (p2) at (4,1.25) { $P_{h}^{2}$ }; \node (p3) at (6,1.25) { $P_{h}^{3}$ }; \draw[->] (p0) -- node[above] { $+ τ_{1} p_{h}^{1}$ } (p1); \draw[->] (p1) -- node[above] { $+ τ_{2} p_{h}^{2}$ } (p2); \draw[->] (p2) -- node[above] { $+ τ_{3} p_{h}^{3}$ } (p3); \draw[->] (0,0) -- (6.4,0) node[right] { $t$ }; \foreach \x/\lab in {0/ $t_{0}$ ,2/ $t_{1}$ ,4/ $t_{2}$ ,6/ $t_{3}$ } \draw (\x,0.08) -- (\x,-0.08) node[below] {\lab}; \node at (1,0.38) { ${\overset{ˉ}{p}}_{h} = p_{h}^{1}$ }; \node at (3,0.38) { ${\overset{ˉ}{p}}_{h} = p_{h}^{2}$ }; \node at (5,0.38) { ${\overset{ˉ}{p}}_{h} = p_{h}^{3}$ }; \end{tikzpicture}

Example references

smooth MINI observation interface

accumulated-pressure observation bound

Guermond, Chapter 54, §54.2

Boffi, Brezzi, and Fortin, §§8.4.2 and 8.6.1

Facts & Assumptions

Given: Countable Choice, a smooth MINI interface on the unit square with three time cells, a nonconstant $\hat{η} \in C_{c}^{\infty} ((\frac{1}{4}, \frac{3}{4}))$ , and the sensor and normalized time test in the Example.

[A1]

Countable Choice means that every family $(X_{n})_{n \in N}$ of nonempty sets has a choice function $f$ with $f (n) \in X_{n}$ . (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The accumulated-pressure proposition bounds the endpoint observation by $K H ∥ ξ ∥_{Q}$ and the time-tested observation by $(K + 1) H ∥ η^{'} ∥_{L^{1}} ∥ ξ ∥_{Q}$ under the stated smooth-branch hypotheses. (Accumulated pressure-observation error for the smooth MINI branch)

[F2]

The interface defines the accumulated-pressure observation $(P (t_{m}) - P_{h}^{m}, ξ)_{L^{2}}$ and the time-tested pressure observation ${(\int_{0}^{T} (p (t) - {\overset{ˉ}{p}}_{h} (t)) η (t) d t, ξ)}_{L^{2}}$ . (Smooth MINI observation interface)

[L1]

Direct integration on the unit square gives $\int_{D} (x - \frac{1}{2}) d x d y = 0$ and $\int_{D} (x - \frac{1}{2})^{2} d x d y = 1 / 12$ .

[L2]

If the smooth compactly supported function $\hat{η}$ were to have $∥ {\hat{η}}^{'} ∥_{L^{1}} = 0$ , then continuity of its derivative would force ${\hat{η}}^{'} = 0$ everywhere, making $\hat{η}$ constant; this contradicts the given nonconstancy.

Verification

Given: The interface and tests in the Example.

1.1givenL1algebra

By [L1], $ξ \in Q$ and $∥ ξ ∥_{Q} = 1 / \sqrt{12}$ .

1.2givenL2algebra

By [L2], $∥ {\hat{η}}^{'} ∥_{L^{1} (0, 1)} > 0$ , so $η$ is defined and has $∥ η^{'} ∥_{L^{1} (0, 1)} = 1$ .

2.1A1F1F2step 1.1step 1.2algebra∎

Apply [F1] using the interface observations in [F2] with this $ξ$ and $η$ . Substituting the two norms from steps 1.1 and 1.2 gives the endpoint bound $K H / \sqrt{12}$ at every $0 \leq m \leq 3$ and the time-tested bound $(K + 1) H / \sqrt{12}$ . These are conditional mathematical error bounds, not measured values.

A mean-zero square pressure time test

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources