We present a systematic construction of finite element exact sequences with a commuting diagram property 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.
Bibliographical noteFunding Information:
∗Received by the editors May 2, 2016; accepted for publication (in revised form) February 9, 2017; published electronically July 13, 2017. http://www.siam.org/journals/sinum/55-4/M107335.html Funding: The work of the first author was supported in part by the National Science Foundation (grant DMS-1522657) and by the University of Minnesota Supercomputing Institute. †School of Mathematics, University of Minnesota, Minneapolis, MN 55455 (cockburn@math. umn.edu). ‡Division of Applied Mathematics, Brown University, Providence, RI 02912 (guosheng fu@ brown.edu).
© 2017 Society for Industrial and Applied Mathematics.
- Commuting diagrams
- Exact sequences
- Finite elements
- Polyhedral elements