Interpolation by spline spaces on classes of triangulations
This paper presents an inductive method for the construction of triangulations \(\Delta\). These triangular elements are suitable for interpolation by the space \(S^r_q(\Delta)\) of splines of degree \(q\) and smoothness \(r= 1,2\). The article is divided into seven sections. The procedures to construct triangulations and admissible sets are described in sections 2 and 3, respectively. The notion of admissible sets, which is considered here for technical reasons, is given in terms of the Bézier-Bernstein representation \[ p^{[l]}(x, y)= \sum_{i+j+k= q} a^{[l]}_{i,j,k} {q!\over i!j!k!} \Phi^i_1(x,y) \Phi^j_2(x, y)\Phi^k_3(x, y),\quad (x,y)\in T^{[l]}, \] where \(p^{[l]}= s|_{T^{[l]}}\) are the polynomials pieces of the spline \(s\in S^r_q(\Delta)\), \(T^{[l]}= \Delta(v^{[l]}_1, v^{[l]}_2, v^{[l]}_3)\) are the triangles of \(\Delta\), \(l= 1,\dots, N\), and \(\Phi_\mu\), \(\mu= 1,2,3\), are linear polynomials uniquely defined by \(\Phi_\mu(v^{[l]}_\nu)= \delta_{\mu,\nu}\), \(\nu= 1,2,3\). Some sets \({\mathcal A}_r\subset I= \{(i,j,k,l): i+ j+ k= q, l= 1,\dots, N\}\), are found to be admissible sets for \(S^r_q(\Delta)\), \(r= 1,2\). In this respect two theorems are presented in the third section for the cases \(q\geq 3\), \(r= 1\); and \(q\geq 5\), \(r= 2\). The problem of constructing Lagrange and Hermite interpolation sets for \(S^r_q(\Delta)\) is investigated in section 4. Sections 5 and 6 are devoted to prove the main theorems of the paper and contain other results of technical interest. In order to illustrate the power of the new method, the last section contains tables with numerical results on interpolation of functions and scattered data. The procedure is used to approximate the test function of Franke, by \(S^1_q(\Delta)\), \(q= 3,4\), and \(S^1_2(\Delta)\).
scientific article; zbMATH DE number 1491265
scientific article