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

Sablonniere, Paul

J. Comput. Appl. Math. 12/13, 541-550 (1985).

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

Keywords:

Hermite problem

References:

[1]	Barnhill, R.E.; Farin, G., C1 quintic interpolation over triangles: two explicit representations, Internat. J. numer. methods engrg., 17, 1763-1778, (1981) · Zbl 0477.65009
[2]	Bézier, P., Mathematical and practical possibilities of UNISURF, (), 127-152
[3]	Bézier, P., Essai de definition numérique des courbes et des surfaces expérimentales, () · Zbl 0157.42201
[4]	Birkhoff, G.; Mansfield, L., Compatible triangular finite elements, J. math. anal. appl., 47, 531-553, (1974) · Zbl 0284.35021
[5]	Ciarlet, P.G., Sur l’élément de clough et tocher, RAIRO rech. opér., 2, 19-27, (1974) · Zbl 0306.65070
[6]	Ciarlet, P.G., Interpolation error estimates for the reduced hsieh-clough-tocher triangle, Math. comput., 32, 142, 335-344, (1978) · Zbl 0378.65010
[7]	Clough, R.W.; Tocher, J.L., Finite element stiffness matrices for analysis of plates in bending, ()
[8]	Farin, G., Subsplines über dreiecken, Dissertation, (1979), Braunschweig
[9]	Farin, G., Bézier polynomials over triangles and the construction of piecewise C^{r}-polynomials, ()
[10]	Mansfield, L., Higher order compatible triangular finite elements, Numer. math., 22, 89-97, (1974) · Zbl 0265.65011
[11]	Le Mehaute, A., Taylor interpolation of order n at the vertices of a triangle. applications for Hermite interpolation and finite elements, (), 171-185
[12]	Le Mehaute, A., Interpolation et approximation par des fonctions polynomiales par morceaux dans \(R\)^{n}, ()
[13]	Powell, M.J.D.; Sabin, M.A., Piecewise quadratic approximations on triangles, ACM trans. math. software, 3, 4, 316-325, (1977) · Zbl 0375.41010
[14]	Sablonniere, P., Interpolation d’Hermite par des surfaces de classe C1 quadratiques par morceaux, (), 175-185 · Zbl 0453.41002
[15]	Sablonniere, P., Bases de Bernstein et approximants splines, ()
[16]	Sablonniere, P., Error bounds for Hermite interpolation by quadratic splines on an α-triangulation, (), IMA J. numer. anal., (1984), submitted
[17]	P. Sablonniere, Triangular finite elements in \(P\)^{2}5(T, τ6), Rapport ANO, Université de Lille I, to appear.
[18]	P. Sablonniere, Triangular finite elements in \(P\)^{2}7(T, τ3), Rapport ANO, Université de Lille I, to appear.
[19]	Ženišek, A., Interpolation polynomials on the triangle, Numer. math., 15, 283-296, (1970) · Zbl 0216.38901
[20]	Ženišek, A., A general theorem on triangular finite C^{m}-elements, RAIRO anal. numer., 2, 119-127, (1974) · Zbl 0321.41003