Quasi-optimal complexity $hp$-FEM for the Poisson Equation on a rectangle

TL;DR

This work develops a provably quasi-optimal $hp$-FEM solver for the (screened) Poisson equation on rectangles by exploiting a sparse $B^3$-Arrowhead discretisation derived from integrated Legendre functions. A reverse Cholesky factorisation is shown to have linear-time complexity and can be parallelised across elements, enabling $O(N^2 \log N)$ setup/solve in 2D, while fast Legendre transforms enable quasi-optimal time-evolution for nonlinear PDEs. The framework is extended to general boundary conditions and recast PDEs into Sylvester or generalized Sylvester equations, which are then solved efficiently using a generalized ADI method with proven spectral bounds. The method also supports preconditioning of variable-coefficient PDEs and uses efficient transforms to move between coefficient and grid representations, demonstrated on Burgers’ equation and singular-coefficient problems, with potential for scalable parallel and higher-dimensional implementations.

Abstract

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 à 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.

Figures (7)

• Figure 1: Time taken to build/factorise and solve a discretisation of a 1D (screened) Poisson equation up to degree $p$, where $n$ is the number of elements. The $x$-axis is $N = np-1$, the total number of degrees of freedom and demonstrates that the complexity is optimal as either $n\ (= 2/h)$ or $p$ become large, and largely only depends on the total number of degrees of freedom. \newlabelFigure:onedtims0
• Figure 1: Example PDE with a discontinuous right-hand side $f$ (left) and solution (right) of the screened Poisson equation $-\Delta u+10^2 u = f$ with a zero Dirichlet boundary condition. By using a $9 \times 9$ elements we can resolve the right-hand side exactly, and then use high $p$ to achieve convergence. \newlabelFigure:discont0
• Figure 1: Left: Solution of Burgers' equation $u_t + u u_x = \epsilon \Delta u$ with a zero Dirichlet boundary condition, where the initial condition is the discontinuous right-hand side as in Figure \ref{['Figure:discont']}, with $\epsilon = 0.1$. Right: time taken for a single time-step, using an implicit-explicit splitting method where the linear part is solved using implicit Euler and the nonlinear part with explicit Euler, after transforming back to a grid. \newlabelFigure:burgers0
• Figure 1: Left: Graded tensor-product mesh \ref{['eq:graded-mesh']} with $m=3$. Right: Solution of singular variable coefficient problem \ref{['eq:variable-coeff']} with partial (pre-tensor) degree $p=128$ and graded mesh \ref{['eq:graded-mesh']} with $m=3$.
• Figure 2: Time taken to factorise and solve a 1D (screened) Poisson equation up to degree $p$, where $n= 8$ is the number of elements, where we parallelise over multiple cores, on an M2 MacBook Air with 4 high performance and 4 high efficiency cores, using shared memory. We see substantial speedup for factorisation when utilising the 4 high performance cores whilst a marginal speedup when also using the high efficiency cores. \newlabelFigure:onedtims_multithread0

Theorems & Definitions (16)

• Remark 1.1
• Definition 4.1
• Theorem 4.2
• Proof 1
• Corollary 4.3
• Proof 2
• Remark 4.4
• Lemma 5.1
• Proof 3
• Lemma 5.2: Spectrum