FoCM 2014 conference

Workshop C2 - Foundation of Numerical PDE's

December 20, 14:30 ~ 15:05 - Room B21

## Quadrilateral $Q_k$ elements and the regular decomposition property

### Gabriel Acosta

### University of Buenos Aires, Argentina - gacosta@dm.uba.ar

Let $K\subset \mathbb{R}^2$ be a convex quadrilateral. In [1] the following definition can be found: $K$ satisfies the regular decomposition property with constants $N<\infty$ and $0<\psi<\pi$ if we can divide $K$ into two triangles along one of its diagonals, called $d_1$, in such a way that $|d_2|/|d_1|< N$ and the maximum angle of both triangles is bounded by $\psi$. Moreover, in [1] it is shown that the constant in the estimate of the $H_1$ norm of the error for the $Q_1$-Lagrange interpolation can be bounded in terms of $N$ and $\psi$. In [2] this result is generalized to $W^{1,p}$ for $1\le p<3$, while for $3\le p$ it is shown that the constant can be bounded in terms of the minimal and the maximal angle of $K$. In this talk we show the role of the regular decomposition property in quadrilateral $Q_k$ interpolation for $k\ge 2$.

[1] G. Acosta, R. Duran Error estimates for $Q_1$-isoparametric elements satisfying a weak angle condition. SIAM J. Numer. Anal. 38, 1073-1088, 2000.

[2] G. Acosta, G. Monzon Interpolation error estimates in $W^{1,p}$ for degenerate $Q_1$-isoparametric elements. Numer. Math. , 104, pp 129-150, 2006.

Joint work with Gabriel Monzón (Universidad de General Sarmiento, Argentina).