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Fortin stability and quasi-optimality for practical DPG

Statement

Assume Countable Choice and the real form and spaces of Ultraweak Poisson DPG spaces and plan-consistent bilinear form on a finite conforming triangular mesh. For every admissible local input to Exact local finite Riesz solves for practical DPG tests, the exact factorization terminates after $O (N_{K}^{3})$ scalar arithmetic operations once its Gram entries and right-hand side are formed; it returns singular-test-matrix exactly for a dependent input family, and on an independent family it returns the unique finite Riesz representative. Let $U_{h} \subset U$ consist of volume fields $σ_{h} ∣_{K} \in P_{1} (K)^{2}$ , $u_{h} ∣_{K} \in P_{1} (K)$ , continuous scalar traces ${\hat{u}}_{h} ∣_{\partial K} \in {\tilde{P}}_{2} (\partial K)$ , and conservative normal traces ${\hat{σ}}_{n, h} ∣_{\partial K} \in P_{1} (\partial K)$ . This is finite-dimensional. Let $V_{r} = \prod_{K} (P_{3} (K)^{2} \times P_{3} (K))$ , presented cellwise by fixed ordered bases, and let $T_{r} : U_{h} \to V_{r}$ be the assembled finite Riesz map. On each fixed basis the unique local Riesz representatives assemble to a bounded linear map satisfying $∥ T_{r} w_{h} ∥_{V} \leq ∥ b ∥ ∥ w_{h} ∥_{U}$ . Now assume the continuous primal and adjoint graph inf-sup constants have minimum $β > 0$ . Let $Π : V \to V_{r}$ be the bounded Fortin map of A uniformly bounded Fortin map for the practical P1/R3 DPG family, with $b (U_{h}, (I - Π) V) = 0$ and $∥ Π ∥ \leq C_{Π}$ . Then for every bounded load $F \in V^{'}$ , the practical DPG system tested with $T_{r} (U_{h})$ has a unique solution $x_{h}$ , and $∥ x - x_{h} ∥_{U} \leq (1 + ∥ b ∥ C_{Π} / β) \inf_{z_{h} \in U_{h}} ∥ x - z_{h} ∥_{U},$ where $x$ is the exact continuous solution.

Facts & Assumptions

Given: Countable Choice, the real DPG form and spaces on a finite triangular mesh, admissible local finite-Riesz inputs, the stated P1/R3 trial and test spaces, the two continuous inf-sup hypotheses, and the specified Fortin map.

[A1]

Countable Choice is used by the mixed BNB solve. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The DPG spaces and bilinear form are those of the ultraweak definition. (Ultraweak Poisson DPG spaces and plan-consistent bilinear form)

[F2]

The continuous form is bounded. (The ultraweak Poisson DPG bilinear form is bounded)

[F3]

The practical Fortin map is bounded and preserves the form on $U_{h}$ . (A uniformly bounded Fortin map for the practical P1/R3 DPG family)

[F4]

The BNB theorem gives a unique mixed solution when the constraint inf-sup holds and the first form is coercive on its kernel. (Babuska-Brezzi solvability by kernel coercivity and inf-sup)

[F5]

Riesz represents the bounded form and datum on the real Hilbert trial and test spaces; the adjoint operator is the Hilbert adjoint. (Riesz representation for Hilbert spaces)

[F6]

The practical test map $T_{r}$ is the finite Riesz representative in the broken polynomial test space. (Exact local finite Riesz solves for practical DPG tests)

[F7]

The trial and test Hilbert factors are complete: real L2 and H1 are Hilbert, H(div) is Hilbert, and the trace quotient factors are Hilbert. Finite products preserve completeness. ( $L^{2}$ with the integral pairing is a Hilbert space, The Sobolev space $H^{1}$ is a Hilbert space, H(div) is Hilbert and H0(div) is closed, Ultraweak Poisson skeleton spaces are Hilbert trace quotients)

Proof

1.1F2F5F6givenalgebra

On an arbitrary local input family, its Gram matrix is positive semidefinite. As long as the preceding vectors are independent, the next unpivoted $L D L^{T}$ pivot is the squared norm of that vector's residual after orthogonal projection onto their span; this follows by solving the preceding Gram equations. Hence every pivot is positive exactly while the vectors remain independent, and the first dependent vector gives a zero pivot, so the algorithm returns singular-test-matrix exactly for a dependent family. If the family is independent, the finite loop and triangular solves terminate, the resulting coefficients solve $G c = g$ , and the returned vector is therefore the unique finite Riesz representative. The factorization and solves use $O (N_{K}^{3})$ exact scalar operations after entry formation. For each cell this representative satisfies $(T_{r, K} w_{h}, v_{K})_{V (K)} = b_{K} (w_{h}, v_{K})$ ; summing over cells gives $(T_{r} w_{h}, v_{r})_{V} = b (w_{h}, v_{r})$ for every $v_{r} \in V_{r}$ . For the fixed bases of $V_{r}$ , local uniqueness and linearity of $b$ show that the assembled $T_{r}$ is linear. Its Riesz identity and [F2] give $∥ T_{r} w_{h} ∥_{V} \leq ∥ b ∥ ∥ w_{h} ∥_{U}$ .

1.2F1F2F5F7givenalgebra

By [F7], $U, V$ are real Hilbert spaces; boundedness [F2] and Riesz [F5] define $A : U \to V$ by $(A w, v)_{V} = b (w, v)$ . The primal inf-sup gives $∥ A w ∥_{V} \geq β ∥ w ∥_{U}$ , so $A$ is injective and its range is closed: if $A w_{n}$ converges, the lower bound makes $w_{n}$ Cauchy, and its limit maps to the range limit. The adjoint inf-sup gives $∥ A^{*} v ∥_{U} \geq β ∥ v ∥_{V}$ , so $\ker A^{*} = {0}$ . By the defining adjoint identity, $v ⊥ ran (A)$ iff $(A w, v)_{V} = 0$ for every $w$ , iff $(w, A^{*} v)_{U} = 0$ for every $w$ , iff $A^{*} v = 0$ . Hence $ran (A)^{⊥} = \ker A^{*}$ , the closed range is dense, and therefore all of $V$ . Riesz represents $F \in V^{'}$ , so it has a unique preimage $x \in U$ .

1.3F3givenalgebra

For $w_{h} \in U_{h}$ , the continuous primal bound gives $β ∥ w_{h} ∥_{U} \leq \sup_{v \in V} b (w_{h}, v) / ∥ v ∥_{V}$ . Fortin orthogonality in [F3] gives $b (w_{h}, v) = b (w_{h}, Π v)$ and $∥ Π v ∥_{V} \leq C_{Π} ∥ v ∥_{V}$ , so the restricted inf-sup on $V_{r}$ is at least $γ = β / C_{Π}$ .

2.1A1F2F4F6step 1.3

On $V_{r} \times U_{h}$ , put $a (r, v) = (r, v)_{V}$ and $c (v, w_{h}) = b (w_{h}, v)$ . The first form is coercive on all of $V_{r}$ , and [F4]’s constraint inf-sup is $γ$ by step 1.3. Applying [F4] with datum $F ∣_{V_{r}}$ and zero constraint gives a unique pair $(r_{h}, x_{h})$ with $(r_{h}, v) + b (x_{h}, v) = F (v)$ and $b (w_{h}, r_{h}) = 0$ . The first equation makes $r_{h}$ the Riesz representative of the restricted residual; the second is orthogonality to $T_{r} (U_{h})$ , exactly the practical DPG equations by [F6].

3.1F2F5step 1.2step 1.3step 2.1algebra∎

Give $V_{r}^{'}$ the inner product transported from $V_{r}$ by its Riesz map. The second mixed equation makes $F - B_{r} x_{h}$ orthogonal to every $B_{r} w_{h}$ in this inner product. For any $z_{h}$ , $F - B_{r} z_{h} = (F - B_{r} x_{h}) + B_{r} (x_{h} - z_{h})$ , so Pythagoras gives $∥ B_{r} (x_{h} - z_{h}) ∥_{V_{r}^{'}} \leq ∥ F - B_{r} z_{h} ∥_{V_{r}^{'}} = ∥ B_{r} (x - z_{h}) ∥_{V_{r}^{'}} \leq ∥ b ∥ ∥ x - z_{h} ∥_{U}$ . Apply the discrete inf-sup $γ$ to $x_{h} - z_{h}$ , then the triangle inequality and infimize over $z_{h}$ to obtain the displayed estimate.

Depends on

Used by

Cited to discharge well-definedness by Exact local finite Riesz solves for practical DPG tests.

Dependency tree · two levels

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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)