$H^1$, $H(\mathrm {curl})$ 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.

A Matlab code for poly-cone

We present a 100-line Matlab code for evaluating the vector proxies of Whitney 0-, 1-, and 2-forms, as well as their exterior derivatives, at given points in a poly-cone, i.e., a cone with polygonal base. The code also contains a subroutine which computes vector proxies of 2D Whitney forms and their exterior derivatives on the base polygon, where the Wachspress coordinates are used to define the 0-forms. The code requires Matlab version R2016b or higher, since it uses the ‘implicit expansion’ feature.

The code requires that the input poly-cone must have its base on the xy-plane and apex at [0,0,a] where a$=\theta$ is the height of the poly-cone. Note that any random poly-cone can be easily converted to such one by a simple rotation and shifting. We emphasize that rotation and shifting do not alter the definition of degrees of freedom for H(div) and H(curl) elements. This is different from the case of a general affine transformation, in which the Piola transformation must be used to get the correct shape functions.

In the code, [x, y, z] stands for the coordinates of points to evaluate the Whitney forms, which are usually the Gaussian points. We do not have Gaussian quadrature on a poly-cone. However, one can easily divide the poly-cone into simplices and then use the Gaussian quadrature on sub-simplices to do numerical integration on the poly-cone. [xs, ys, zs] stands for , as defined in (9). The subroutine baseShapes calculates the 2D Whitney forms on the base polygon, evaluated at points [xs, ys]. The subroutine pullback computes the scaled pullback operator $\varPi ^*$ of 0-, 1-, and 2-forms.