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Ideal ultraweak Poisson DPG is the energy best approximation

Statement

Assume Countable Choice and the real trial/test spaces and bilinear form of Ultraweak Poisson DPG spaces and plan-consistent bilinear form, with $∣ b (w, v) ∣ \leq 5 ∥ w ∥_{U} ∥ v ∥_{V}$ as in The ultraweak Poisson DPG bilinear form is bounded. The ideal trial-to-test map $T$ of Ideal Poisson DPG trial-to-test Riesz map is well defined, bounded linear, and satisfies $∥ T ∥ \leq 5$ . Let $x \in U$ be any exact ultraweak solution and let $U_{h} \subset U$ be finite dimensional. The ideal DPG equations with test space $T (U_{h})$ have a unique solution $x_{h}$ , which is the orthogonal projection of $x$ in the induced energy inner product $(w, z)_{E} = (T w, T z)_{V}$ . Consequently $∥ x - x_{h} ∥_{E} = \inf_{z_{h} \in U_{h}} ∥ x - z_{h} ∥_{E} = ∥ B (x - x_{h}) ∥_{V^{'}}$ .

Facts & Assumptions

Given: Countable Choice, the real Hilbert trial/test spaces and bounded bilinear form, an exact solution $x$ , and a finite-dimensional $U_{h}$ .

[A1]

Countable Choice is declared for Hilbert Riesz maps. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The trial operator has trivial kernel. (The ultraweak Poisson trial operator has trivial kernel)

[F2]

The ideal procedure defines $T w$ as the Riesz representative of $b (w, \cdot)$ . (Ideal Poisson DPG trial-to-test Riesz map)

[F3]

Riesz represents bounded real functionals uniquely and identifies each dual norm with the norm of its representative. (Riesz representation for Hilbert spaces)

[F4]

The DPG bilinear form is bounded by $∣ b (w, v) ∣ \leq 5 ∥ w ∥_{U} ∥ v ∥_{V}$ . (The ultraweak Poisson DPG bilinear form is bounded)

Proof

1.1A1F1F2F3F4givenalgebra

For every $w \in U$ , [F4] makes $v \mapsto b (w, v)$ a bounded functional on $V$ , and [F3] gives the unique $T w$ in [F2], with $∥ T w ∥_{V} \leq 5 ∥ w ∥_{U}$ . Bilinearity of $b$ and uniqueness of the Riesz representative imply $T (α w_{1} + β w_{2}) = α T w_{1} + β T w_{2}$ for real scalars $α, β$ , so $T$ is bounded linear. By [F1], $T w = 0$ implies $w = 0$ ; hence $(w, z)_{E} = (T w, T z)_{V}$ is a positive-definite inner product. For every $w_{h} \in U_{h}$ , the ideal equation is $(T x_{h}, T w_{h})_{V} = F (T w_{h}) = (T x, T w_{h})_{V}$ , by [F2] and exactness.

2.1F1step 1.1algebra

Choose an orthonormal basis $e_{1}, \dots, e_{N}$ of the finite-dimensional image $T (U_{h})$ and define $P (T x) = \sum_{j} (T x, e_{j})_{V} e_{j}$ . The equations in step 1.1 identify $T x_{h} = P (T x)$ ; Pythagoras proves it uniquely minimizes distance to that image. Injectivity of $T$ gives the unique $x_{h}$ and the best-approximation identity.

3.1F2F3step 2.1algebra∎

The functional $B (x - x_{h}) = b (x - x_{h}, \cdot)$ has Riesz representative $T (x - x_{h})$ by [F2]. Therefore [F3] gives $∥ B (x - x_{h}) ∥_{V^{'}} = ∥ T (x - x_{h}) ∥_{V} = ∥ x - x_{h} ∥_{E}$ .

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Cited to discharge well-definedness by Ideal Poisson DPG trial-to-test Riesz map.

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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)