Composite finite elements of class \(C^ k\) (Q1072031)

Let \(\tau\) be a triangulation of \(D\subset {\mathbb{R}}^ 2\) with vertices \(A=\{A_ i,1\leq i\leq N\}\) and \({\mathbb{P}}^ k_ n(D,\tau)=\{\nu \in C^ k(D):\forall T\in \tau\), \(v| T\in {\mathbb{P}}_ n\}\) where \({\mathbb{P}}_ n:=\{polynomials\) of total degree \(\leq n\}\). For \(u\in C^ r(D)\), \(r\geq k\), and V finite dimensional subspace of \(C^ k(D)\), we consider the Hermite problem \(H^ k(A,V):=\{find\) \(v\in V:D^{\alpha}v(A_ i)=D^{\alpha}u(A_ i)\), \(1\leq i\leq N,| \alpha | \leq k\}\). This problem is solvable in \(V={\mathbb{P}}^ k_ n(D,\tau)\) iff \(n\geq 4k+1\) (Ženišek). When \(V={\mathbb{P}}^ k_ n(D,\tau_ 3)\), where \(\tau_ 3\) is a subtriangulation of \(\tau\) obtained by subdividing each \(T\in \tau\) into 3 triangles, the problem has a solution for \(n=4k-1\) (e.g. the \(C^ 1\)-cubic HCT triangle and a \({\mathbb{P}}^ 2_ 7\)-triangle with 31 parameters). When \(V={\mathbb{P}}^ k_ n(D,\tau_ 6)\), where \(\tau_ 6\) is a subtriangulation of \(\tau\) obtained by subdividing each \(T\in \tau\) into 6 triangles, the problem has a solution for \(n=3k-1\) (e.g. the \(C^ 1\)-quadratic PS triangle and a \({\mathbb{P}}^ 2_ 5\)- triangle with 31 parameters). In both cases, the construction needs the partial derivatives \(D^{\alpha}u(A_ i)\) for \(| \alpha | \leq 2k-1\). The domain D is assumed to be polygonal, compact and connected.