A systematic construction of finite element commuting exact sequences
For researchers in finite element methods, this work provides a unified framework to generate exact sequences, but the results are largely incremental as they extend existing constructions to new element types.
This paper presents a systematic construction of finite element exact sequences with commuting diagrams for the de Rham complex in 1D, 2D, and 3D. The construction rediscovers known families and uncovers several new families of exact sequences for various element types, including polygons, pyramids, prisms, and cubes.
We present a systematic construction of finite element exact sequences with a commuting diagram for the de Rham complex in one-, two- and three-space dimensions. We apply the construction in two-space dimensions to rediscover two families of exact sequences for triangles and three for squares, and to uncover one new family of exact sequence for squares and two new families of exact sequences for general polygonal elements. We apply the construction in three-space dimensions to rediscover two families of exact sequences for tetrahedra, three for cubes, and one for prisms; and to uncover four new families of exact sequences for pyramids, three for prisms, and one for cubes.