In the master's thesis we consider local construction of polynomial trivariate splines over a tetrahedral partition $\triangle$. For the representation of trivariate polynomials of degree $n$ over a tetrahedron $T$ we use the Bernstein basis and connect it to its set of domain points $\mathcal{D}_{n,T}$. We take a look at efficient and stable computation of derivatives at the vertices, on the edges, on the faces and in the interior of tetrahedra using De Casteljau algorithm and polynomial blossoms. We further introduce three $C^1$ superspline spaces over tetrahedral partition $\triangle$, its Alfeld refinement $\triangle_{\rm{A}}$ and Worsey-Farin refinement $\triangle_{\rm{WF}}$ and find their nodal minimal determining sets $\mathcal{N}$, $\mathcal{N}_{\rm{A}}$ and $\mathcal{N}_{\rm{WF}}$. Consequently, these spaces are indeed $C^1$ macroelement spaces. They are used for finding the solutions of Hermite interpolation problems defined by $\mathcal{N}$, $\mathcal{N}_{\rm{A}}$ and $\mathcal{N}_{\rm{WF}}$.
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