Quasi-optimal complexity $hp$-FEM for Poisson on a rectangle
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 th...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
17.02.2024
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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DOI: | 10.48550/arxiv.2402.11299 |