This is a brief note on the choice of Gauss-Lobatto-Legendre (GLL) points and a summary of degree counting. GLL points are widely used in spectral element methods (SEMs), but initially, I only relied on their definition and did not fully connect them to the Gauss nodes, which are standard textbook material. It wasn’t until I TAed CS450 that I was forced to finally had the chance to sit down and think it through.
Legendre Polynomials
With the unweighted inner product defined as $(f,g)=\int_{-1}^1 f(x)g(x)dx$, the Legendre polynomials, denoted as $L_N(x)$ with degree $N$, can be constructed via the orthogonalization of the monomials basis \(\{1,x,x^2,\cdots,x^N\}\). Therefore, we have:
\[\int_{-1}^1L_N(x)\ x^j\ dx = 0, \quad\quad\forall j=0,\cdots,N-1\]It can be shown (homework 9, CS450, FA24) that all Legendre polynomials have simple root within the open interval $(-1,1)$. Using these roots, denoted as \(\{\xi^{GL}_i\}_{i=1,\cdots,N}\), as the nodes, we can define the Lagrange polynomials $\ell^{GL}_i(x)$ and construct the Gauss-Legendre (GL) quadrature:
\[\int_{-1}^1 f(x)\ dx \approx \sum_{i=1}^N w^{GL}_i f(\xi^{GL}_i)\quad {\rm where\ } w^{GL}_i:= \int_{-1}^1\ell^{GL}_i(x)\ dx,\]Also, it can be shown (Part B, hw 9) that the quadrature rule is exact for any polynomial $f$ of degree up to $2N-1$, using the polynomial remainder theorem and the orthogonality relation.
Gauss-Lobatto
To include the endpoints, which is necessary to represent the boundary conditions for SEM, the nodes must contain $\pm 1$. Therefore, the $n$ collocation points are the roots of \(p_n(x) = (x^2-1)q_{n-2}(x)\) where \(q_{n-2}\in\mathbb{P}_{n-2}\) is some polynomial of degree $n-2$. Here we examine two possible choices for $q$: $q_{n-2}=L_{n-2}$ or $q_{n-2}=L’_{n-1}$.
Case $q_{n-2}=L_{n-2}$
Due to the extra order introduced by $x^2-1$, the orthogonality relations
\[\int_{-1}^1 p_n(x)\ x^j\ dx = \int_{-1}^1 L_{n-2}(x)\ (x^{j+2}-x^j)\ dx = 0\]hold only for $j+2\leq n-3$, or $j=0,\cdots,n-5$.
Case $q_{n-2}=L’_{n-1}$
By applying the integration by part, we obtain
\[\begin{align*} 0 &= \int_{-1}^1 p_n(x)\ x^j\ dx = \int_{-1}^1 (x^2-1) x^j L'_{n-1}(x)\ dx = \int_{-1}^1 (x^2-1) x^j \ d L_{n-1}(x)\\ &= \cancelto{0}{\left[(x^2-1) x^j L_{n-1}(x)\right]\bigg|_{-1}^1} - \int_{-1}^1 L_{n-1}(x) \left((j+2)x^{j+1}-jx^{j-1}\right)\ dx. \end{align*}\]This relaxes two extra orders and the equation holds when $j+1\leq n-2$ or $j=0,\cdots, n-3$.
The second choice leverages more orthogonality provided by the Legendre polynomials. The Gauss-Lobatto-Legendre (GLL) nodes \(\{\xi^{GLL}_i\}_{i=1,\cdots,n}\) are then defined as the roots of \((x^2-1)L'_{n-1}(x)\) and the quadrature rule is exact for polynomials up to degree $2n-3$.
Comparison
In summary, for degree $N$ Legendre polynomials $L_N(x)$ and $n$ collocation points, the following table summarizes the quadrature rule of Legendre family. Here, the Gauss-Radau-Legendre (GRL) points are for the semi-open interval that includes $-1$, which is convenient to avoid singularity of the radial axis under the polar coordinate.
| Method | Roots of | Range | Orthogonality | Quad. Exact Deg. |
|---|---|---|---|---|
| Gauss | $L_N(x)$ | $(-1,1)$ | $n-1$ | $2n-1$ |
| Gauss-Radau | $L_N(x)+L_{N-1}(x)$ | $[-1,1)$ | $n-2$ | $2n-2$ |
| Gauss-Lobatto | $(x^2-1)L’_N(x)$ | $[-1,1]$ | $n-3$ | $2n-3 = 2N-1$ |