template<class D, class R, int k>
class Dune::LagrangePyramidLocalFiniteElement< D, R, k >
Lagrange finite element for 3d pyramids with compile-time polynomial order.
- Template Parameters
-
D | Type used for domain coordinates |
R | Type used for shape function values |
k | Polynomial order, only orders 1 and 2 are actually implemented |
Lagrange shape functions are tricky. In the paper mentioned below, Christian Wieners states "There exists no continuously differentiable conforming shape function for the pyramid
which is linear, resp. bilinear on the faces." The same holds, mutatis mutandis, for second-order Lagrange functions. The usual remedy, employed here, is to use shape functions that are continuous, but only piecewise differentiable.
More specifically, the implementations in this file are taken from the following papers:
- First order: C. Wieners, Conforming discretizations on tetrahedrons, pyramids, prisms and hexahedrons (1997)
- Second order: L. Liu et.al. On Higher Order Pyramidal Finite Elements, (2011) [anisotropic variant]
These are piecewise trilinear/triquadratic basis functions with the following properties:
- Shape functions with Lagrange property
- Each basis function is bilinear/biquadratic on the rectangular face
- Each basis function is linear/quadratic on all triangular faces
- Each basis function is continuous in the interelement boundary
As the derivatives are not continuous, numerical quadrature for expressions involving shape function derivatives should employ a composite rule composed of two tetrahedral parts.
- Warning
- The shape functions currently do not sum up to 1, even though my understanding of the Liu et al. paper is that they should.