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 1/|J|(p-1) for 1 <= p <= infinity on the reference element, where J is the Jacobian of the nona. ne 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 is an element of [1, infinity), 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 not equal 2. 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 <= p <= infinity.