## Composite finite elements of class $$C^ k$$.(English)Zbl 0587.41004

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.

### MSC:

 41A05 Interpolation in approximation theory 41A10 Approximation by polynomials 41A30 Approximation by other special function classes

Hermite problem
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### References:

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