Constructions of some minimal finite element systems
This work provides a theoretical foundation for designing minimal finite element spaces, which is relevant for researchers in numerical analysis and computational engineering.
The paper presents a unified framework for constructing minimal finite element systems that contain prescribed functions and have commuting interpolators, demonstrating that known spaces like trimmed polynomial forms, serendipity elements, and TNT elements satisfy this design principle. A dimension formula for minimal systems on hypercubes is also provided.
Within the framework of finite element systems, we show how spaces of differential forms may be constructed, in such a way that they are equipped with commuting interpolators and contain prescribed functions, and are minimal under these constraints. We show how various known mixed finite element spaces fulfill such a design principle, including trimmed polynomial differential forms, serendipity elements and TNT elements. We also comment on virtual element methods and provide a dimension formula for minimal compatible finite element systems containing polynomials of a given degree on hypercubes.