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 reverse Cholesky factorisation (Cholesky starting from the bottom right instead of the top left) remains sparse. Moreover, computing and inverting the factorisation may parallelise across 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 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. We also demonstrate how the solver can be used as an effective preconditioner for PDEs with variable coefficients, including coefficients that support a singularity.

翻译：我们在一维情形下证明，$hp$-有限元方法（$hp$-FEM）离散化可在最优复杂度内求解，因为该离散化具有特殊的稀疏结构，确保反向Cholesky分解（从右下角而非左上角开始的Cholesky分解）保持稀疏性。此外，计算和求逆该分解可在不同单元间并行执行。通过将这一方法融入Fortunato与Townsend（2020）提出的交替方向隐式（ADI）算法框架，我们能够在预设容差内以拟最优复杂度求解矩形域上（屏蔽）泊松方程的$hp$-FEM离散化：计算复杂度为 $O(N^2 \log N)$，其中 $N$ 为各维度最大总自由度。结合快速勒让德变换，我们还能以 $O(N^2 \log^2 N)$ 的拟最优复杂度求解非线性时间演化偏微分方程，并以（粘性）Burgers方程为例进行验证。我们同时展示了该求解器如何作为含变系数（包括存在奇异性的系数）偏微分方程的有效预处理器。