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Solvers for mixed finite element problems using Poincar\'e operators based on spanning trees
Numerical Analysis · Submitted 11 Oct 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We propose a decomposition of Hilbert complexes that directly leads to a Poincar\'e operator.
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The abstract, sentence by sentence
We propose a decomposition of Hilbert complexes that directly leads to a Poincar\'e operator. An explicit example is provided that decomposes a finite element differential complex using spanning trees in the grid. The Poincar\'e operator has three implications. First, it yields a new basis in which the mixed formulation of the Hodge-Laplace problem unravels from a large saddle point system into seven smaller, symmetric positive definite systems. These systems can be solved sequentially, and three of these have the same dimensions as the cohomology classes. Second, we use the operator to construct an explicit basis for the harmonic forms. Third, we propose an auxiliary space preconditioner for problems in weighted Sobolev spaces, that robustly handles the large kernel of the differential operator. These three implications are validated through numerical experiments.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose a decomposition of Hilbert complexes that directly leads to a Poincar\'e operator. | contribution 0.95 | 0.93 | 0.23 | asserted 1.00 |
| s1 | An explicit example is provided that decomposes a finite element differential complex using spanning trees in the grid. | method 0.58 | 0.92 | 0.07 | asserted 1.00 |
| s2 | The Poincar\'e operator has three implications. | interpretation 0.70 | 0.67 | 0.08 | asserted 0.99 |
| s3 | First, it yields a new basis in which the mixed formulation of the Hodge-Laplace problem unravels from a large saddle point system into seven smaller, symmetric positive definite systems. | result 0.48 | 0.90 | 0.30 | asserted 1.00 |
| s4 | These systems can be solved sequentially, and three of these have the same dimensions as the cohomology classes. | result 0.71 | 0.82 | 0.09 | asserted 1.00 |
| s5 | Second, we use the operator to construct an explicit basis for the harmonic forms. | contribution 0.81 | 0.94 | 0.09 | asserted 1.00 |
| s6 | Third, we propose an auxiliary space preconditioner for problems in weighted Sobolev spaces, that robustly handles the large kernel of the differential operator. | contribution 0.89 | 0.93 | 0.14 | asserted 1.00 |
| s7 | These three implications are validated through numerical experiments. | result 0.77 | 0.91 | 0.19 | asserted 1.00 |
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Original abstract
We propose a decomposition of Hilbert complexes that directly leads to a Poincar\'e operator. An explicit example is provided that decomposes a finite element differential complex using spanning trees in the grid. The Poincar\'e operator has three implications. First, it yields a new basis in which the mixed formulation of the Hodge-Laplace problem unravels from a large saddle point system into seven smaller, symmetric positive definite systems. These systems can be solved sequentially, and three of these have the same dimensions as the cohomology classes. Second, we use the operator to construct an explicit basis for the harmonic forms. Third, we propose an auxiliary space preconditioner for problems in weighted Sobolev spaces, that robustly handles the large kernel of the differential operator. These three implications are validated through numerical experiments.
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This is possible down to the lowest order Nédélec elements in the $hp$-finite element method using space-splitting methods [ref] , or even fully with non-local approaches [ref] .
Cosserat micropolar and couple-stress elasticity models of flexomagnetism at finite deformations · Discretisation · 0.81
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