Z.: On the interpolation error estimates for Q1 quadrilateral finite elements

Abstract. 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 1|J|p−1 for 1 ≤ p ≤ ∞ on the reference element, where J is the Jacobian of the non-affine mapping. This allows us to introduce weak geometric conditions leading to interpolation error estimates in W 1,p norm, conditions that can be regarded as a general-ization of the RDP (regular decomposition property) condition introduced in [2]. We avoid the use of the reference family elements, that allows us to extend the results to a larger class of elements and to introduce the GRDP condition in a more unified way. As far as we know, the GRDP condition presented in this paper is weaker than any other mesh conditions proposed in the literatures for any p with 1 ≤ p ≤ ∞. Key words. Error estimates, quadrilateral elements, isoparametric finite elements, maximal angle condition AMS subject classification. 65N30, 65N1