Gridap.Polynomials

This module provides a collection of multivariate polynomial bases.

The exported names are:

• MonomialBasis
• QCurlGradMonomialBasis
• QGradMonomialBasis
• change_basis
• get_exponents
• get_order
• get_orders
• get_value_type
• num_terms

struct MonomialBasis{D,T} <: Field

Type representing a basis of multivariate scalar-valued, vector-valued, or tensor-valued, iso- or aniso-tropic monomials. The fields of this struct are not public This type fully implements the Field interface, with up to second order derivatives.

MonomialBasis{D}(::Type{T}, order::Int [, filter::Function]) where {D,T}

Returns an instance of MonomialBasis representing a multivariate polynomial basis in D dimensions, of polynomial degree order, whose value is represented by the type T. The type T is typically <:Number, e.g., Float64 for scalar-valued functions and VectorValue{D,Float64} for vector-valued ones.

Filter function

The filter function is used to select which terms of the tensor product space of order order in D dimensions are to be used. If the filter is not provided, the full tensor-product space is used by default leading to a multivariate polynomial space of type Q. The signature of the filter function is

(e,order) -> Bool

where e is a tuple of D integers containing the exponents of a multivariate monomial. The following filters are used to select well known polynomial spaces

• Q space: (e,order) -> true
• P space: (e,order) -> sum(e) <= order
• "Serendipity" space: (e,order) -> sum( [ i for i in e if i>1 ] ) <= order

MonomialBasis{D}(::Type{T}, orders::Tuple [, filter::Function]) where {D,T}

This version of the constructor allows to pass a tuple orders containing the polynomial order to be used in each of the D dimensions in order to construct and anisotropic tensor-product space.

struct QCurlGradMonomialBasis{...} <: Field

This type implements a multivariate vector-valued polynomial basis spanning the space needed for Raviart-Thomas reference elements on n-cubes. The type parameters and fields of this struct are not public. This type fully implements the Field interface, with up to first order derivatives.

QCurlGradMonomialBasis{D}(::Type{T},order::Int) where {D,T}

Returns a QCurlGradMonomialBasis object. D is the dimension of the coordinate space and T is the type of the components in the vector-value.

struct QGradMonomialBasis{...} <: Field

This type implements a multivariate vector-valued polynomial basis spanning the space needed for Nedelec reference elements on n-cubes. The type parameters and fields of this struct are not public. This type fully implements the Field interface, with up to first order derivatives.

QGradMonomialBasis{D}(::Type{T},order::Int) where {D,T}

Returns a QGradMonomialBasis object. D is the dimension of the coordinate space and T is the type of the components in the vector-value.

change_basis(basis,changeofbasis::AbstractMatrix)

Examples

Compute the Lagrangian basis associated with a set of nodes

using LinearAlgebra
using Gridap.Fields
using Gridap.Polynomials

D = 2
order = 1
f = MonomialBasis{D}(Float64,order)

nodes = Point{2,Int}[(0,0),(1,0),(0,1),(1,1)]
change = inv(evaluate(f,nodes))

g = change_basis(f,change)
println(evaluate(g,nodes))

# output
[1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0]

get_exponents(b::MonomialBasis)

Get a vector of tuples with the exponents of all the terms in the monomial basis.

Examples

using Gridap.Polynomials

b = MonomialBasis{2}(Float64,2)

exponents = get_exponents(b)

println(exponents)

# output
Tuple{Int64,Int64}[(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (2, 2)]

get_order(b::MonomialBasis)

get_orders(b::MonomialBasis)

num_terms(f::QCurlGradMonomialBasis{D,T}) where {D,T}