  We show, in one dimension, that an $hp$-Finite Element Method ($hp$-FEM)
discretisation can be solved in optimal complexity because the discretisation
has a special sparsity structure that ensures that the \emph{reverse Cholesky
factorisation} -- Cholesky starting from the bottom right instead of the top
left -- remains sparse. Moreover, computing and inverting the factorisation
almost entirely trivially parallelises across the different elements. By
incorporating this approach into an Alternating Direction Implicit (ADI) method
\`a la Fortunato and Townsend (2020) we can solve, within a prescribed
tolerance, an $hp$-FEM discretisation of the (screened) Poisson equation on a
rectangle, in parallel, with quasi-optimal complexity: $O(N^2 \log N)$
operations where $N$ is the maximal total degrees of freedom in each dimension.
When combined with fast Legendre transforms we can also solve nonlinear
time-evolution partial differential equations in a quasi-optimal complexity of
$O(N^2 \log^2 N)$ operations, which we demonstrate on the (viscid) Burgers'
equation.
