Nonstandard finite element de Rham complexes on cubical meshes
For computational scientists and engineers, this provides a systematic way to design finite element methods with reduced complexity and improved regularity for solving higher-order PDEs.
The paper introduces two operations (DoF-transfer and serendipity) on finite element de Rham complexes for cubical meshes, enabling construction of nonstandard elements (Hermite, Adini, trimmed-Adini) that reduce degrees of freedom while increasing continuity. These elements yield convergent, non-conforming methods satisfying a discrete Korn inequality, with potential applications to Stokes, biharmonic, and elasticity problems.
We propose two general operations on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of "nonstandard" convergent methods. The first operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reduces global degrees of freedom while increasing continuity order at vertices. The second operation, called serendipity, eliminates interior bubble functions and degrees of freedom locally on each element without affecting edge degrees of freedom. These operations can be used independently or in tandem to create nonstandard complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show that the resulting elements provide convergent, non-conforming methods for problems requiring stronger regularity and satisfy a discrete Korn inequality. We discuss potential benefits of applying these elements to Stokes, biharmonic and elasticity problems.