Subdivision and spline spaces

TL;DR

This paper investigates how mesh refinement affects spline spaces on simplicial or polyhedral meshes, providing conditions for direct sum decompositions, dimension formulas, and explicit bases for various subdivisions.

Contribution

It introduces sufficient conditions for spline module decompositions under mesh subdivision and derives explicit bases and dimension formulas for common subdivisions.

Findings

Spline modules split as direct sums under certain subdivisions.

Explicit bases are constructed for various mesh refinements.

Dimension formulas are established for spline spaces on subdivided meshes.

Abstract

A standard construction in approximation theory is mesh refinement. For a simplicial or polyhedral mesh D in R^k, we study the subdivision D' obtained by subdividing a maximal cell of D. We give sufficient conditions for the module of splines on D' to split as the direct sum of splines on D and splines on the subdivided cell. As a consequence, we obtain dimension formulas and explicit bases for several commonly used subdivisions and their multivariate generalizations.