Error estimation of anisotropic interpolation for serendipity elements of arbitrary degree

Abstract

Serendipity elements are a class of simplified forms of the double-k element in 2D or the triple-k element in 3D. The main advantage is that serendipity elements can maintain a proper order of convergence of the interpolation error under the condition of regularity, while reducing the number of inner freedoms. In practice, it is necessary to analyze their interpolation error on anisotropic meshes. Based on quasi-orthogonal polynomials, the explicit forms of interpolating polynomials for 2D and 3D serendipity elements with arbitrary degrees are first presented. Then, the interpolation error estimates of the serendipity elements on anisotropic meshes are given. Furthermore, the fine estimation of constants appearing in the interpolation estimates is derived. Finally, numerical experiments are performed for specific serendipity elements and the numerical results show the correctness of the theoretical analysis.