The mixed-order serendipity finite element for H(curl)-conforming hexahedra

A new serendipity function space for hexahedral H(curl)-conforming finite elements, the so-called mixed-order serendipity space, is proposed. In the case of arbitrary parallelepiped meshes, the resulting asymptotic rate of convergence of the H(curl)-norm error is exponential in the base of the mesh size and in terms of the finite element order. Compared to tensor product spaces, the number of unknowns is much smaller, whereas the rate of convergence remains the same. While the proposed serendipity space is not suitable for elements of general shape, it allows the construction of hierarchical basis functions that are compatible with some of the costly but versatile tensor product spaces. This property permits mixing different finite element spaces within one mesh, without affecting conformity.For the general curvilinear case, this paper introduces an iso-serendipity finite element for curvilinear hexahedra, which employs the proposed serendipity space and basis for representing the fields and H
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serendipity basis for representing the geometry. As the main contribution, this element achieves the same convergence rate for both curvilinear and parallelepiped meshes, by means of a special yet simple mesh refinement technique. In contrast to isogeometric methods, it only requires certain interpolation points on curvilinear boundaries rather than the entire geometry mapping. All results are supported by mathematical proofs and validated experimentally via numerical examples.