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A constant Darcy flux on the reference triangle

Statement

Assume Countable Choice.

On $K = {(x, y) : x > 0, y > 0, x + y < 1}$ with unit permeability, $q = (1, 0)$ and $p = - x$ satisfy $q + \nabla p = 0$ and $div q = 0$ . Their outward integrated facet fluxes on $x = 0$ , $y = 0$ , and $x + y = 1$ are respectively $- 1, 0, + 1$ , so exact local mass balance holds.

Facts & Assumptions

Given: The reference triangle, unit permeability, the constant flux $q = (1, 0)$ , and affine pressure $p = - x$ .

[A1]

Countable Choice is inherited from the mesh and RT0 unisolvence interfaces. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

RT0 facet fluxes determine the local RT0 field, and its divergence is constant. (Affine simplicial meshes and RT/BDM spaces, RT0 facet-flux unisolvence and global H(div) conformity)

[F2]

The Gauss–Green formula equates the divergence integral with the sum of outward facet fluxes. (Gauss–Green formula on an affine simplex)

Proof

1.1givenalgebra

The gradient is $\nabla p = (- 1, 0)$ , so $q + \nabla p = 0$ pointwise. The three outward unit normals are $(- 1, 0)$ on $x = 0$ , $(0, - 1)$ on $y = 0$ , and $(1, 1) / \sqrt{2}$ on the hypotenuse. Their facet lengths are $1, 1, \sqrt{2}$ , respectively. Therefore the integrated fluxes are $- 1$ , $0$ , and $(1 / \sqrt{2}) \sqrt{2} = 1$ .

2.1A1F1F2step 1.1algebra∎

Their sum is zero, equal to $\int_{K} div q = 0$ by [F2]. Since $q$ is in RT0 and the fluxes are its facet degrees of freedom by [F1], the data also uniquely determine this constant flux among RT0 fields.

Mesh diagram

The triangle is the set $x > 0, y > 0, x + y < 1$ . The blue arrows indicate the outward-normal directions; their magnitudes are schematic. The proof computes the exact integrated fluxes, so the drawing supplies orientation only.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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

Francisco-Javier Sayas, A gentle introduction to the Finite Element Method (2015) (standard reference, not scraped)
Jean-Luc Guermond, MATH 661 Fall 2025, Part III Chapter 14: H(div) finite elements (standard reference, not scraped)