The dimenstion and bases of divergence-free splines; a homological approach

For a triangulated 2-dimensional region in \(\mathbb{R}^2 \) , let S′_m(Δ) be the vector space of all C′ functions F on Δ such that for any simplex σ∈Δ, F|σ is a polynomial of degree at most m. Let D′_m(Δ) be the vector space consisting of all pairs (F₁, F₂) with F_i∈S′_m(Δ), such that Σ_t(?F_t/?x_t=0, i.e., the pair is divergence-free. Both S′_m(Δ) and D′_m(Δ) can be described in terms of chain complexes using the usual boundary map of homology, and these complexes can be related by an epimorphism. When Δ is 2-disk the epimorphism gives the explicit result that \(\mathbb{R}^2 \) . Bases for D′_m(Δ) are derived from bases of S′ _m+1 ⁺¹ (Δ) via the epimorphism in this case.