Composite finite elements of class Ck

摘要

Let τ be a triangulation of D ⊂R2 with vertices ≔≔A = {Ai, 1 ⩽ i ⩽ N}andPkn(D, τ) ≔ {v ϵ Ck(D): ∀T ϵ τ, v|T ϵPn} wherePn≔ {polynomials of total degree⩽ n}. For u ϵ Cr(D), r ⩾ k, and V finite dimensional subspace of Ck(D), we consider the Hermite problem Hk(A, V) ≔ {find v ϵ V: Dαv (Ai) = Dαu(Ai), 1 ⩽ i ⩽ N, |α| ⩽ k}. This problem is solvable in V =Pkn(D, τ) iff n ⩾ 4k+ 1 (Ženišek). When V =Pkn(D, τ3), where τ3 is a subtriangulation of τ obtained by subdividing each T ϵ τ into 3 triangles, the problem has a solution for n = 4k − 1. (e.g. the C1-cubic HCT triangle and a P27-triangle with 31 parameters). When V =Pkn(D, τ6), where τ6 is a subtriangulation of τ obtained by subdividing each T ϵ τ into 6 triangles, the problem has a solution for n = 3k − 1 (e.g. the C1-quadratic PS triangle and a P25-triangle with 31 parameters). In both cases, the construction needs the partial derivatives Dαu(Ai) for |α| ⩽ 2k − 1. The domain D is assumed to be polygonal, compact and connected.