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The degree-two GLL mass matrix changes only its highest Legendre mode

Example

For $p = 2$ , the one-dimensional Legendre basis $(P_{0}, P_{1}, P_{2})$ has exact mass matrix $diag (2, 2 / 3, 2 / 5)$ , whereas the GLL quadrature mass matrix is $diag (2, 2 / 3, 1)$ . The only change is the top mode, with correction $3 / 5$ .

Facts & Assumptions

Given: $p = 2$ , $P_{0} = 1$ , $P_{1} = x$ , $P_{2} = (3 x^{2} - 1) / 2$ , and the GLL definition.

[F1]

Legendre-Gauss-Lobatto nodes, weights, and tensor quadrature: the $p = 2$ nodes are $- 1, 0, 1$ with weights defined by integrating their cardinal polynomials.

[F2]

GLL quadrature changes only the top Legendre modes and is p-uniformly equivalent: for degree $p$ , the GLL squared-norm defect is $2 (p + 1) / (p (2 p + 1))$ times the squared top Legendre coefficient.

[F3]

Legendre polynomials have explicit norms and weighted derivative orthogonality: $\int_{- 1}^{1} P_{n} P_{m} = 2 δ_{n m} / (2 n + 1)$ .

Verification

1.1F1givenalgebra

Since $P_{2}^{'} (x) = 3 x$ , its interior GLL node is $0$ , and the two endpoints complete the nodes. The cardinal polynomials are $ℓ_{- 1} = x (x - 1) / 2$ , $ℓ_{0} = 1 - x^{2}$ , and $ℓ_{1} = x (x + 1) / 2$ . Integrating them gives weights $1 / 3, 4 / 3, 1 / 3$ .

2.1F2F3step 1.1givenalgebra∎

Legendre orthogonality [F3] gives the exact diagonal entries $\int P_{0}^{2} = 2$ , $\int P_{1}^{2} = 2 / 3$ , and $\int P_{2}^{2} = 2 / 5$ . The nodes and weights in step 1.1 give $Q_{2} (P_{0}^{2}) = 2$ , $Q_{2} (P_{1}^{2}) = 2 / 3$ , and $Q_{2} (P_{2}^{2}) = 1$ . Every cross product has degree at most three, so exactness through degree $2 p - 1 = 3$ makes the off-diagonal entries zero. The top correction is $1 - 2 / 5 = 3 / 5$ , which agrees with [F2] at $p = 2$ and top coefficient $1$ .

Depends on

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Sources

Lloyd N. Trefethen, Short note on the mass matrix for Gauss-Lobatto grid points (standard reference, not scraped)
Lucas Johannes Petrus Timmermans, Analysis of Spectral Element Methods with Application to Incompressible Flow (standard reference, not scraped)