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Adjoint graph inf-sup for ultraweak Poisson on connected Lipschitz polyhedra

Statement

Assume Countable Choice. For every bounded connected Lipschitz polyhedron $Ω \subset R^{d}$ , $d \in {2, 3}$ , and every finite conforming shape-regular simplicial mesh, give $U$ its $L^{2}$ /global-quotient product norm and $V$ the full broken graph norm of Ultraweak Poisson DPG spaces and plan-consistent bilinear form. With $C_{*}$ from A topology-free bounded lifting of every ultraweak Poisson load, the primal and adjoint bounds are $\sup_{y \neq 0} \frac{∣ b (x, y) ∣}{∥ y ∥_{V}} \geq C_{*}^{- 1} ∥ x ∥_{U}, \sup_{x \neq 0} \frac{∣ b (x, y) ∣}{∥ x ∥_{U}} \geq C_{*}^{- 1} ∥ y ∥_{V} .$ Both constants are independent of the mesh. The bounded operator $B : U \to V^{'}$ and its Riesz trial-to-test map $T : U \to V$ are boundedly invertible, with inverse norms at most $C_{*}$ . No simple-connectivity or connected-boundary hypothesis is required; connected polyhedra with holes are included.

Facts & Assumptions

Given: Countable Choice, the stated connected Lipschitz polyhedron and finite mesh, and the exact real trial/test norms and bilinear form.

[A1]

Countable Choice is the hypothesis for the Hilbert and Riesz interfaces used below. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The product trial and broken test spaces are real Hilbert spaces: their $L^{2}$ factors, global trace quotient ranges, $H (div)$ factors and $H^{1}$ factors have the stated real Hilbert structures. ( $L^{2}$ with the integral pairing is a Hilbert space, Ultraweak Poisson skeleton spaces are Hilbert trace quotients, H(div) is Hilbert and H0(div) is closed, The Sobolev space $H^{1}$ is a Hilbert space)

[F2]

The real bilinear form defines a bounded operator $B : U \to V^{'}$ with $∣ b (x, y) ∣ \leq 5 ∥ x ∥_{U} ∥ y ∥_{V}$ . (The ultraweak Poisson DPG bilinear form is bounded)

[F3]

The trial operator has trivial kernel: $B x = 0$ implies $x = 0$ . (The ultraweak Poisson trial operator has trivial kernel)

[F4]

There is a linear $R : V^{'} \to U$ with $B R = I_{V^{'}}$ and $∥ R ∥ \leq C_{*}$ . (A topology-free bounded lifting of every ultraweak Poisson load)

[F5]

The real Hilbert Riesz map $J_{V} : V \to V^{'}$ is an isometric linear bijection. (Riesz representation for Hilbert spaces)

[F6]

The formulation fixes the five signs of $b$ , the real trial/test spaces and the full broken graph norm. (Ultraweak Poisson DPG spaces and plan-consistent bilinear form)

Proof

1.1A1F2F3F4F6givenalgebra

The bounded form [F2] defines $B$ , and [F4] makes $B$ onto. By [F3], $B$ is injective. For each $x \in U$ , $B (R B x - x) = B R B x - B x = 0$ , so injectivity gives $R B x = x$ . Therefore $B^{- 1} = R$ , $∥ x ∥_{U} \leq C_{*} ∥ B x ∥_{V^{'}}$ , and the dual norm identity $∥ B x ∥_{V^{'}} = \sup_{y \neq 0} ∣ b (x, y) ∣ / ∥ y ∥_{V}$ gives the primal bound.

2.1F1F4F5step 1.1algebra

For $y \neq 0$ , set $F = J_{V} y$ and $x = R F$ . Then $B x = F$ , so $b (x, y) = F (y) = ∥ y ∥_{V}^{2}$ , while [F4] and [F5] give $∥ x ∥_{U} \leq C_{*} ∥ F ∥_{V^{'}} = C_{*} ∥ y ∥_{V}$ . Since $F \neq 0$ , $x \neq 0$ , and this $x$ proves the adjoint bound; for $y = 0$ the bound is immediate.

3.1F1F2F4F5F6step 1.1step 2.1algebra∎

Define $T = J_{V}^{- 1} B$ . By [F2], $∥ T ∥ \leq 5$ . Since $B^{- 1} = R$ , $T^{- 1} = R J_{V}$ and [F4] and [F5] give $∥ T^{- 1} ∥ \leq C_{*}$ . The lifting constant depends only on a fixed-domain Poincaré constant, so both graph inf-sup bounds are uniform over the stated mesh family; the construction uses no simple-connectivity or connected-boundary condition.

Depends on

Used by

P1/R3 practical DPG quasi-optimality and residual estimator for Poisson Corollary

Dependency tree · two levels

53 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

Jay Gopalakrishnan and Weifeng Qiu, An analysis of the practical DPG method (standard reference, not scraped)
Leszek Demkowicz and Jay Gopalakrishnan, Analysis of the DPG method for the Poisson equation (standard reference, not scraped)
Leszek Demkowicz, Mathematical Theory of Finite Elements (standard reference, not scraped)
Carstensen, Demkowicz, and Gopalakrishnan, A Posteriori Error Control for DPG Methods (standard reference, not scraped)