Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Weak and pointwise divergence-free discrete velocities
Definition
For the conforming mini-element spaces , define the weak discrete divergence kernel by and the exact discrete divergence kernel by Since , one has . Equality is not part of either definition; the companion example proves strictness for a concrete mesh.
Depends on
Used by
- Graded MINI mass initialization can destroy the smooth instantaneous pressure rate Counterexample
- The mixed mini-element backward Euler Navier–Stokes scheme Definition
- A weakly divergence-free mini velocity need not be pointwise divergence-free Example
- Second-order L2 approximation by the MINI Fortin map Lemma
- Strong L2 compactness for weakly solenoidal MINI trajectories Lemma
- Well-definedness of the finite-dimensional mini backward Euler equations Lemma
- Approximation by the MINI solenoidal energy projection Theorem
- Backward Euler MINI step existence and discrete energy bound Theorem
- Discrete Helmholtz projection and pressure Schur identities Theorem
- MINI backward Euler converges subsequentially to a solenoidal energy weak velocity Theorem
- Smooth-branch velocity error for backward Euler MINI Theorem
Dependency tree · two levels
11 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
- Jean-Luc Guermond, MATH 661 Fall 2025, Chapters 49–54 (complete course notes) (standard reference, not scraped)
- Daniele Boffi, Franco Brezzi, Michel Fortin, Mixed Finite Element Methods and Applications (complete textbook) (standard reference, not scraped)