Short note on the mass matrix for Gauss-Lobatto grid points

TL;DR

This paper demonstrates that the exact mass matrix and its inverse for Gauss-Lobatto points differ from their diagonal approximations by a simple rank-1 update, enabling efficient application in $O(N)$ operations.

Contribution

It reveals that the exact mass matrix and inverse differ from approximations by a rank-1 update, allowing efficient computation.

Findings

Exact mass matrix differs from diagonal approximation by a rank-1 update.

Inverse of the mass matrix can be applied in $O(N)$ operations.

Simplifies computations in spectral methods using Gauss-Lobatto points.

Abstract

The mass matrix for Gauss-Lobatto grid points is usually approximated by Gauss-Lobatto quadrature because this leads to a diagonal matrix that is easy to invert. The exact mass matrix and its inverse are full. We show that the exact mass matrix \emph{and} its inverse differ from the approximate diagonal ones by a simple rank-1 update (outer product). They can thus be applied to an arbitrary vector in $O(N)$ operations instead of $O(N^2)$.