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

Journal article

We show, in one dimension, that an -Finite Element Method (-FEM) discretization can be solved in optimal complexity because the discretization has a special sparsity structure that ensures that the reverse Cholesky factorization—Cholesky starting from the bottom right instead of the top left—remains sparse. Moreover, computing and inverting the factorization may parallelize across different elements. By incorporating this approach into an Alternating Direction Implicit method (Fortunato D. and Townsend A. (2020) Fast Poisson solvers for spectral methods. IMA J. Numer. Anal., 40, 1994–2018) we can solve, within a prescribed tolerance, an -FEM discretization of the (screened) Poisson equation on a rectangle with quasi-optimal complexity: operations where 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 (PDEs) in a quasi-optimal complexity of 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.

Authors: Knook, K; Olver, S; Papadopoulos, IPA

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Oxford University Press
• Journal: IMA Journal of Numerical Analysis
• Article number: draf102
• Publication date: 2025-11-17
• DOI: 10.1093/imanum/draf102
• EISSN: 1464-3642
• ISSN: 0272-4979
• Language: English
• Keywords: quasi-optimal complexity; hp-FEM; Alternating Direction Implicit method; fast direct solvers