Interpolation error estimates of a modified 8-node serendipity finite element

Summary.

Interpolation error estimates for a modified 8-node serendipity finite element are derived in both regular and degenerate cases, the latter of which includes the case when the element is of triangular shape. For $u \in W^{3, p}(K)$ defined over a quadrilateral K, the error for the interpolant $\Pi_K u$ is estimated as $|u-\Pi_K u|_{W^{\alpha, p}(K)}\le Ch^{3-\alpha}_K|u|_{W^{3,p}(K)}$ $(\alpha=0, 1)$, where $1 \le p \le +\infty$ in the regular case and $1\le p < 3$ in the degenerate case, respectively. Thus, the obtained error estimate in the degenerate case is of the same quality as in the regular case at least for $1\le p<3$. Results for some related elements are also given.