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.

翻译：我们证明，在一维情形下，$hp$-有限元法（$hp$-FEM）离散化可以在最优复杂度下求解，因为该离散化具有特殊的稀疏结构，确保了\textit{反向乔列斯基分解}（从右下角而非左上角开始的乔列斯基分解）保持稀疏性。此外，因子分解的计算与求逆几乎完全可在不同单元间平凡并行化。通过将该方法与Fortunato和Townsend（2020）提出的交替方向隐式（ADI）方法相结合，我们能够在给定容差内，以拟最优复杂度$O(N^2 \log N)$（其中$N$为每个维度的最大总自由度）并行求解矩形域上（屏蔽）泊松方程的$hp$-FEM离散化。结合快速勒让德变换，我们还可以$O(N^2 \log^2 N)$的拟最优复杂度求解非线性时间演化偏微分方程，并在（黏性）伯格斯方程上进行了验证。