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The ultraweak Poisson DPG bilinear form is bounded
Statement
Assume Countable Choice. On the real trial and broken graph-test spaces of Ultraweak Poisson DPG spaces and plan-consistent bilinear form, is bounded by , uniformly in mesh size.
Facts & Assumptions
Given: Product graph and skeleton quotient norms.
Countable Choice is declared for trace duality. (The Axiom of Countable Choice ())
The five form terms and signs are fixed by the DPG definition. (Ultraweak Poisson DPG spaces and plan-consistent bilinear form)
Skeleton norms are global lift infima. (Ultraweak Poisson skeleton spaces are Hilbert trace quotients)
Weak normal trace is bounded in H(div). (Weak normal trace by Green duality)
Proof
Cauchy–Schwarz bounds the three volume pairings in [F1] by the trial/test product norms.
Lift to global H1_0 and use [F3] on . Lift to global H(div) and use cell Green identities on broken . Cauchy–Schwarz across cells and infimizing by [F2] bound each skeleton term by the same product.
Each of five terms is bounded by the product norm by steps 1.1 and 2.1, so the triangle inequality gives constant , independent of the mesh.
Depends on
Used by
- Ideal Poisson DPG trial-to-test Riesz map Algorithm
- P1/R3 practical DPG quasi-optimality and residual estimator for Poisson Corollary
- A topology-free bounded lifting of every ultraweak Poisson load Lemma
- Adjoint graph inf-sup for ultraweak Poisson on connected Lipschitz polyhedra Theorem
- Fortin stability and quasi-optimality for practical DPG Theorem
- Ideal ultraweak Poisson DPG is the energy best approximation Theorem
Dependency tree · two levels
19 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)
- C. Carstensen, L. Demkowicz, and J. Gopalakrishnan, A Posteriori Error Control for DPG Methods (standard reference, not scraped)