Dune::LagrangePyramidLocalFiniteElement< D, R, k > Class Template Reference

Lagrange finite element for 3d pyramids with compile-time polynomial order. More...

#include <dune/localfunctions/lagrange/lagrangepyramid.hh>

Public Types
using	Traits = LocalFiniteElementTraits< Impl::LagrangePyramidLocalBasis< D, R, k >, Impl::LagrangePyramidLocalCoefficients< k >, Impl::LagrangePyramidLocalInterpolation< Impl::LagrangePyramidLocalBasis< D, R, k > > >
	Export number types, dimensions, etc.

Public Member Functions
const Traits::LocalBasisType &	localBasis () const
	Returns the local basis, i.e., the set of shape functions.
const Traits::LocalCoefficientsType &	localCoefficients () const
	Returns the assignment of the degrees of freedom to the element subentities.
const Traits::LocalInterpolationType &	localInterpolation () const
	Returns object that evaluates degrees of freedom.

Static Public Member Functions
static constexpr std::size_t	size ()
	The number of shape functions.
static constexpr GeometryType	type ()
	The reference element that the local finite element is defined on.

Detailed Description

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.

---

The documentation for this class was generated from the following file:

• dune/localfunctions/lagrange/lagrangepyramid.hh