Quasi-optimal complexity hp-FEM for Poisson on a rectangle
CoRR(2024)
摘要
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
almost entirely trivially parallelises across the 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, 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.
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