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An explicit finite model separates ideal and practical test vectors

Statement

Let $U = V = R^{2}$ with Euclidean inner product, $A = (\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array})$ , $b (x, y) = (A x) \cdot y$ , $U_{h} = span (e_{2})$ , practical test space $V_{r} = span (e_{1})$ , and load $F (y) = e_{1} \cdot y$ . Then $T e_{2} = (1, 1)$ , $T^{r} e_{2} = e_{1}$ , and ideal and practical trial coefficients are $1 / 2$ and $1$ .

Facts & Assumptions

Given: The stated finite-dimensional trial/test data.

[F1]

In this finite model, define $T w \in V$ by $(T w, y)_{V} = b (w, y)$ for every $y \in V$ ; existence and uniqueness follow from finite-dimensional Riesz representation. (Finite-dimensional Riesz representation: every functional is uniquely $v \mapsto ⟨ v, w ⟩$ )

[F2]

Define $T^{r} w \in V_{r}$ by $(T^{r} w, y)_{V} = b (w, y)$ for every $y \in V_{r}$ ; finite-dimensional Riesz representation again gives existence and uniqueness. (Finite-dimensional Riesz representation: every functional is uniquely $v \mapsto ⟨ v, w ⟩$ )

[F3]

Finite-dimensional Riesz systems have unique representatives. (Finite-dimensional Riesz representation: every functional is uniquely $v \mapsto ⟨ v, w ⟩$ )

Proof

1.1F1F2givenalgebra

$A e_{2} = (1, 1)$ , so $(A e_{2}, y)_{V} = b (e_{2}, y)$ for every $y \in V$ , and [F1] gives $T e_{2} = (1, 1)$ . For every $y = t e_{1} \in V_{r}$ , $(e_{1}, y)_{V} = t = b (e_{2}, y)$ , so [F2] gives $T^{r} e_{2} = e_{1}$ .

2.1F1F2F3step 1.1algebra∎

For ideal coefficient $c$ , the equation is $b (c e_{2}, T e_{2}) = F (T e_{2})$ , hence $2 c = 1$ and $c = 1 / 2$ . For practical tests it is $b (c e_{2}, e_{1}) = F (e_{1})$ , hence $c = 1$ . Uniqueness follows from the positive Gram scalars $2$ and $1$ in [F3].

Remarks

The Euclidean Riesz maps defined here give a finite-dimensional analogue of Ideal Poisson DPG trial-to-test Riesz map and Exact local finite Riesz solves for practical DPG tests.

Depends on

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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)
Carstensen, Demkowicz, and Gopalakrishnan, A Posteriori Error Control for DPG Methods (standard reference, not scraped)