Second Order Convergence of the Interpolation Based on $Q^c_1$-Element

Abstract

In this paper, the second order convergence of the interpolation based on $Q^c_1$-element is derived in the case of $d$=1, 2 and 3. Using the integral average on each element, the new basis functions of tensor product type is builded up and we can easily extend it to the higher dimensional case. Finally, some numerical tests are made to show the analytical results of the interpolation errors.