$$H^1$$, $$H(\mathrm {curl})$$ and $$H(\mathrm {div})$$ conforming elements on polygon-based prisms and cones

Abstract

The conation and extrusion techniques were proposed by Bossavit (Math Comput Simul 80:1567–1577, 2010) for constructing $$(m+1)$$ -dimensional Whitney forms on prisms/cones from m-dimensional ones defined on the base shape. We combine the conation and extrusion techniques with the 2D polygonal $$H(\mathrm {div})$$ conforming finite element proposed by Chen and Wang (Math Comput 307:2053–2087, 2017), and construct the lowest-order $$H^1$$ , $$H(\mathrm {curl})$$ and $$H(\mathrm {div})$$ conforming elements on polygon-based prisms and cones. The elements have optimal approximation rates. Despite of the relatively sophisticated theoretical analysis, the construction itself is easy to implement. As an example, we provide a 100-line Matlab code for evaluating the shape functions of $$H^1$$ , $$H(\mathrm {curl})$$ and $$H(\mathrm {div})$$ conforming elements as well as their exterior derivatives on polygon-based cones. Note that all convex and some non-convex 3D polyhedra can be divided into polygon-based cones by connecting the vertices with a chosen interior point. Thus our construction also provides composite elements for all such polyhedra.