Orthonormal polynomial basis \( \left\{ P_i(X) \right\} \) in dimensions \( d \) and order \( p
\).
More...
template<int Dims, int Order>
struct pbat::math::polynomial::OrthonormalBasis< Dims, Order >
Orthonormal polynomial basis \( \left\{ P_i(X) \right\} \) in dimensions \( d \) and order \( p
\).
The basis is orthonormal with respect to the inner product
\[ \langle f, g \rangle = \int_{\Omega^d} f(X) g(X) \, d\Omega^d
\]
where \( \Omega^d \) is the reference simplex in dimensions \( d \), e.g.
- the line segment \( 0,1 \) in 1D,
- the triangle \(\begin{pmatrix} 0 \\ 0 \end{pmatrix},
\begin{pmatrix} 1 \\ 0 \end{pmatrix},
\begin{pmatrix} 0 \\ 1 \end{pmatrix}
\) in 2D, and
- the tetrahedron \(\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix},
\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix},
\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix},
\begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix}
\) in 3D.
In other words,
\[ \langle P_i, P_j \rangle = \delta_{ij}
\]
where \( \delta_{ij} \) is the Kronecker delta.
See Orthogonal polynomials.
- Template Parameters
-
| Dims | Spatial dimensions |
| Order | Polynomial order |
1.13.2