Dune::LagrangeSimplexLocalFiniteElement< D, R, d, compileTimeOrder > Class Template Reference

Lagrange finite element for simplices with arbitrary compile-time dimension and static or dynamic polynomial order. More...

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

Public Types
using	Traits = LocalFiniteElementTraits< Impl::LagrangeSimplexLocalBasis< D, R, d, compileTimeOrder >, Impl::LagrangeSimplexLocalCoefficients< d, compileTimeOrder >, Impl::LagrangeSimplexLocalInterpolation< Impl::LagrangeSimplexLocalBasis< D, R, d, compileTimeOrder > > >
	Export number types, dimensions, etc.

Public Member Functions
constexpr	LagrangeSimplexLocalFiniteElement ()
	Constructor for compile-time order.
constexpr	LagrangeSimplexLocalFiniteElement (int runTimeOrder)
	Constructor for run-time order.
template<typename VertexMap > requires (compileTimeOrder >= 0)
constexpr	LagrangeSimplexLocalFiniteElement (const VertexMap &vertexmap)
	Constructs a finite element given a vertex reordering.
constexpr const Traits::LocalBasisType &	localBasis () const
	Returns the local basis, i.e., the set of shape functions.
constexpr const Traits::LocalCoefficientsType &	localCoefficients () const
	Returns the assignment of the degrees of freedom to the element subentities.
constexpr const Traits::LocalInterpolationType &	localInterpolation () const
	Returns object that evaluates degrees of freedom.
constexpr std::size_t	size () const
	The number of shape functions.

Static Public Member Functions
static constexpr GeometryType	type ()
	The reference element that the local finite element is defined on.

Detailed Description

template<class D, class R, int d, int compileTimeOrder = -1>
class Dune::LagrangeSimplexLocalFiniteElement< D, R, d, compileTimeOrder >

Lagrange finite element for simplices with arbitrary compile-time dimension and static or dynamic polynomial order.

Template Parameters

D	type used for domain coordinates
R	type used for function values
d	dimension of the reference element
compileTimeOrder	polynomial order of the shape functions (or -1 for dynamic order)

The Lagrange basis functions \(\phi_i\) of order \(k>1\) on the unit simplex \(G = \{ x \in [0,1]^{d} \,|\, \sum_{j=1}^d x_j \leq 1\}\) are implemented as \(\phi_i(x) = \hat{\phi}_i(kx)\) where \(\hat{\phi}_i\) is the \(i\)-th basis function on the scaled simplex \(\hat{G} = kG = \{ x \in [0,k]^{d} \,|\, \sum_{j=1}^d x_i \leq k\}\). On \(\hat{G}\) the uniform Lagrange points of order \(k\) are exactly the points \((i_1,\dots,i_d) \in \mathbb{N}_0^d \cap \hat{G}\) with non-negative integer coordinates in \(\hat{G}\). Using the lexicographic enumeration of those points we can identify each \(i=(i_1,...,i_d)\) with the flat index \(i\) of the Lagrange node and associated basis function.

Since we can map any point \(\hat{x} \in \hat{G}\) to its barycentric coordinates \((\hat{x}_1, \dots, \hat{x}_d, k-\sum_{j=1}^d \hat{x}_j)\) we define for any such point its auxiliary \((d+1)\)-th coordinate \(\hat{x}_{d+1} = k-\sum_{j=1}^d \hat{x}_j\) and use this in particular for Lagrange points \(i=(i_1,...,i_d)\). Then the Lagrange basis function on \(\hat{G}\) associated to the Lagrange point \(i=(i_1,\dots,i_d)\) is given by

\[ \hat{\phi}_i(\hat{x}) = \prod_{j=1}^{d+1} L_{i_j}(\hat{x}_j) \]

where we used the barycentric coordinates of \(\hat{x}\) and \(i\) and the univariate Lagrange polynomials

\[ L_n(t) = \prod_{m=0}^{n-1} \frac{t-m}{n-m} = \frac{1}{n!}\prod_{m=0}^{n-1}(t-m). \]

This factorization can be interpreted geometrically. To this end note that any component \(1,\dots,d+1\) of the barycentric coordinates is associated to a vertex of the simplex and thus to the opposite facet. Furthermore the Lagrange nodes are all located on hyperplanes parallel to those facets. In the factorized form the term \(L_{i_j}(\hat{x}_j)\) guarantees that \(\hat{\phi}_i\) is zero on any of these hyperplane that is parallel to the facet associated with \(j\) and lying between the Lagrange node \(i\) and this facet (excluding \(i\) and including the facet).

Using the factorized form, evaluating all basis functions \(\hat{\phi}_i\) at a point \(\hat{x}\) requires to first evaluate \(\alpha_{m,j}=L_m(\hat{x}_j)\) for all \(j\in \{1,\dots,d+1\}\) and \(m \in \{0,\dots,k\}\) and then computing the products \(\hat{\phi}_i(\hat{x})=\prod_{j=1}^{d+1} \alpha_{i_j,j}\) for all admissible multi indices (or, equivalently, Lagrange nodes in barycentric coordinates) \((i_1,\dots,i_{d+1})\) with \(\sum_{j=1}^{d+1} i_j = k\). The evaluation of the univariate Lagrange polynomials can be done efficiently using the recursion

\[ L_0(t) = 1, \qquad L_{n+1}(t) = L_n(t)\frac{t-n}{n+1} \qquad n\geq 0. \]

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

• dune/localfunctions/lagrange/lagrangesimplex.hh