Convergence Study of H(curl) Serendipity Basis Functions for Hexahedral Finite-Elements

A new serendipity function space for hexahedral H(curl)-conforming finite-elements, called the mixed-order serendipity space, was recently introduced by the authors. Its key feature is its hierarchical basis. Moreover, the number of basis functions and, consequently, the number of unknowns is significantly less than for the mixed-order Nédélec and tensor product spaces, while the convergence rates are the same. Classical hexahedral finite-element methods may experience a degradation of convergence when meshes of non-parallelpipedal or curvilinear elements are refined. This work presents a numerical convergences study for several H(curl) conforming finite-element bases in the curvilinear case. It is shown that a special, yet versatile, refinement method, called quasi-affine refinement, can restore the optimal rate of convergence in all cases. Amongst the considered finite-element spaces, the mixed-order serendipity spaces require the smallest number of unknowns.