Polynomial bases

The HelFEM.PolynomialBasis type wraps the corresponding C++ class and defines a primitive polynomial basis –- a set of polynomials $\{p_i(x)\}$ on the domain $x \in [-1, 1]$. In the finite element method, these are then repeated in each element.

Currently the library supports three types of polynomials:

:legendre: Legendre polynomials
:hermite: Hermite polynomials
:lip: Lagrange interpolating polynomials (with Gauss-Lobatto nodes)

Each primitive basis is defined by the number of nodes, which then determines the order of the polynomials and how many polynomials are there in the set.

using HelFEM, Plots
xs = range(-1, 1, length=2001)
a = @animate for nnodes = 2:15
    b1 = PolynomialBasis(:legendre, nnodes)
    b2 = PolynomialBasis(:hermite, nnodes)
    b3 = PolynomialBasis(:lip, nnodes)
    plot(
        plot(xs, b1(xs), label=false, title="Legendre, nnodes=$nnodes"),
        plot(xs, b2(xs), label=false, title="Hermite, nnodes=$nnodes"),
        plot(xs, b3(xs), label=false, title="LIP, nnodes=$nnodes"),
        layout = (3, 1), size=(800, 1000)
    )
end
gif(a, fps=0.5)

Various methods are available to work with the polynomial basis and for introspection (see the reference).

Reference

HelFEM.PolynomialBasis — Type

struct PolynomialBasis

Wrapper type for the helfem::atomic::polynomial_basis::PolynomialBasis class, representing a set of polynomials $\{p_i(x)\}$ on a domain $x \in [-1, 1]$.

Constructors

PolynomialBasis(basis::Symbol, nnodes)

Constructs a particular polynomial basis with a specific number of defining nodes. basis must be one of: :hermite, :legendre or :lip.

source

Base.length — Method

length(pb::PolynomialBasis) -> Integer

Returns the number of basis functions (polynomials) represented by the given polynomial basis.

source

HelFEM.nnodes — Method

HelFEM.nnodes(pb::PolynomialBasis) -> Integer

Returns the number of control nodes used to define the polynomial basis in the constructor.

source