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The conforming P1-bubble/P1 mini element
Definition
Assume Countable Choice. Let , or , be a bounded connected Lipschitz polygonal/polyhedral domain with a conforming shape-regular affine simplicial mesh . On a simplex with barycentric coordinates , define the scalar interior bubble . Let The mini-element spaces are Each bubble is extended by zero off its cell. The zero-boundary conformity of the piecewise affine part is supplied by Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare. The local/global dimensions are proved in the next dimension lemma.
Depends on
- Stokes velocity and pressure spaces and weak forms
- Matching affine simplicial meshes and shape regularity
- Broken and conforming polynomial spaces on a matching mesh
- Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare
- Shape regularity controls affine simplex geometry
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
Used by
- Steady MINI Stokes pressure-mass Schur bounds Corollary
- Graded MINI mass initialization can destroy the smooth instantaneous pressure rate Counterexample
- Displacement-pressure formulation and mini-element pair Definition
- The mixed mini-element backward Euler Navier–Stokes scheme Definition
- Weak and pointwise divergence-free discrete velocities Definition
- A fixed-mesh P1 volumetric-locking obstruction Example
- A matching shape-regular MINI mesh family on the unit square Example
- A mini-element bubble detects an affine pressure gradient Example
- A weakly divergence-free mini velocity need not be pointwise divergence-free Example
- Local and global dimensions of the mini element Lemma
- Second-order L2 approximation by the MINI Fortin map Lemma
- The mini-element Fortin map approximates in L2 Lemma
- Well-definedness of the finite-dimensional mini backward Euler equations Lemma
- Discrete Helmholtz projection and pressure Schur identities Theorem
- First-order smooth pressure-primitive error for backward Euler MINI Theorem
- MINI backward Euler converges subsequentially to a solenoidal energy weak velocity Theorem
- Mini-element H1 velocity and L2 pressure error estimate Theorem
- Smooth-branch velocity error for backward Euler MINI Theorem
- Uniform Fortin stability of the mini element Theorem
Dependency tree · two levels
27 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)