On the Interpolation Error Estimates for Q1 Quadrilateral Finite Elements

In this paper, we study the relation between the error estimate of the bilinear interpolation on a general quadrilateral and the geometric characters of the quadrilateral. Some explicit bounds of the interpolation error are obtained based on some sharp estimates of the integral over $\frac{1}{|J|^{p-1}}$ for $1\leq p\leq\infty$ on the reference element, where $J$ is the Jacobian of the nonaffine mapping. This allows us to introduce weak geometric conditions (depending on $p$) leading to interpolation error estimates in the $W^{1,p}$ norm, for any $p\in [1,\infty)$, which can be regarded as a generalization of the regular decomposition property (RDP) condition introduced in [G. Acosta and R. G. Duran, SIAM J. Numer. Anal., 38 (2000), pp. 1073-1088] for $p=2$ and new RDP conditions (NRDP) for $p\neq2$. We avoid the use of the reference family elements, which allows us to extend the results to a larger class of elements and to introduce the NRDP condition in a more unified way. As far as we know, the mesh condition presented in this paper is weaker than any other mesh conditions proposed in the literature for any $p$ with $1\leq p\leq\infty$.

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