$$H^1$$H1 , $$H(\mathrm {curl})$$H(curl) and $$H(\mathrm {div})$$H(div) conforming elements on polygon-based prisms and cones
2020
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.
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