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Nonstandard finite element de Rham complexes on cubical meshes. (English) Zbl 1447.65140
Compatible and stable numerical schemes in the framework of discrete differential forms and finite element exterior calculus are usually constructed discretizing the physical variables in discrete differential complexes, such as finite element de Rham complexes. Nonstandard convergent methods on finite element differential complexes on cubical meshes are derived using degree of freedom (DoF)-transfer operator and serendipity. The DoF-transfer operation moves some edge DoFs to vertex DoFs. The local shape function spaces and bubbles do not change and the DoF transfer preserves unisolvence and exactness. The serendipity operation eliminates some interior bubbles and DoFs at the same time in a way that preserves unisolvence and exactness. In this paper, these two operations are combined to reduce the local and global spaces of known discrete complexes on cubical meshes for the Hermite, Adini and trimmed-Adini type families. Several complexes are constructed with vertex DoFs and fewer local and global DoFs than standard tensor product elements. Potential benefits of these elements applied to problems requiring stronger regularity such as to Stokes, biharmonic and elasticity problems are discussed.
MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65J05 General theory of numerical analysis in abstract spaces
41A15 Spline approximation
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