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Interior facet orientations cancel in a matching mesh
Statement
Orient every top-dimensional cell of a matching simplicial mesh by the ambient orientation of , and orient each facet by the induced boundary convention of Oriented simplicial cell chains and boundary. In the sum of the oriented boundaries of all top-dimensional cells, every interior facet occurs twice with opposite signs; every boundary facet occurs once.
Facts & Assumptions
Given: A finite face-to-face triangulation of a polyhedral domain, with each interior facet incident to two cells and each boundary facet to one.
A matching mesh is face-to-face: a shared facet is a whole facet of both cells. (Matching affine simplicial meshes and shape regularity)
The oriented boundary of an ordered simplex is the alternating sum of its codimension-one faces. (Oriented simplicial cell chains and boundary)
Proof
Let be an interior facet. Its two incident simplex interiors lie on opposite sides of the affine hyperplane containing ; otherwise their interiors would overlap near a relative-interior point of , contradicting the mesh definition. Therefore the two outward unit normals along are opposites.
The boundary orientation is defined by the condition that the outward normal followed by an oriented basis of the facet gives the ambient cell orientation. Replacing the outward normal by its negative reverses the induced facet orientation. Hence the two contributions of in the cell boundary sum have opposite signs and cancel. A boundary facet has one incident cell, so it appears once with its induced boundary orientation.
Depends on
Used by
- Opposite orientations cancel on a two-triangle mesh Example
- A conforming averaging companion for Crouzeix–Raviart functions Lemma
- Element and edge residual representation for P1 Galerkin errors Lemma
- Geometry error bounds a pulled-back boundary-lift residual Lemma
- Weak tangential trace on a Lipschitz polyhedron Lemma
- Well-posedness of the finite FE-HMM micro solves Lemma
- Whitney edge, face, and cell moments commute with grad, curl, and div Lemma
- Continuous matching piecewise polynomials belong to W1p Theorem
- Flux-differencing SBP is conservative and semidiscretely entropy stable Theorem
- Smooth-solution L2 error bound for upwind DG transport Theorem
- Upwind DG transport satisfies the semidiscrete L2 energy identity Theorem
- Zero-boundary conforming polynomials lie in W0 1p and satisfy Poincare Theorem
Dependency tree · one level
2 results within one dependency step 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, Chapter 10, “Mesh orientation” (standard reference, not scraped)
- Joachim Schöberl, Numerical Methods for Partial Differential Equations, Chapter 8 (standard reference, not scraped)