The high-order finite element Duffy de Rham complex and low-order-refined preconditioning
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| Format: | Preprint |
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2026
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| author | Pazner, Will |
| author_facet | Pazner, Will |
| contents | In this work, we construct high-order finite element spaces for the $L^2$ de Rham complex on triangular meshes amenable to low-order-refined preconditioning. The spaces are constructed using the Duffy transformation, by pulling back appropriately chosen polynomial spaces defined on the unit square; in addition to piecewise polynomials, these spaces also contain certain rational functions, and they reduce to the standard Lagrange, Nédélec, and discontinuous finite elements in the lowest-order case. We establish spectral equivalence, independent of the polynomial degree, of the stiffness matrices defined on these spaces with the lowest-order stiffness matrices defined on refined meshes, constructed using a Gauss-Lobatto triangular lattice. Spectral equivalence of the operators is a consequence of norm equivalences in Jacobi-weighted $L^2$ norms, which are established by proving stability of the Jacobi-Gauss-Lobatto interpolation operator in shifted norms. The low-order-refined preconditioners can also be used to precondition the standard piecewise polynomial finite element spaces using a fictitious space approach. The low-order-refined system can in turn be preconditioned effectively using algebraic multigrid methods. The analytical estimates are confirmed by numerical results on a variety of high-order problems, including on mixed meshes and surface meshes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_00148 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The high-order finite element Duffy de Rham complex and low-order-refined preconditioning Pazner, Will Numerical Analysis 65N30, 65F08, 65N55 (Primary) 33C45 (Secondary) In this work, we construct high-order finite element spaces for the $L^2$ de Rham complex on triangular meshes amenable to low-order-refined preconditioning. The spaces are constructed using the Duffy transformation, by pulling back appropriately chosen polynomial spaces defined on the unit square; in addition to piecewise polynomials, these spaces also contain certain rational functions, and they reduce to the standard Lagrange, Nédélec, and discontinuous finite elements in the lowest-order case. We establish spectral equivalence, independent of the polynomial degree, of the stiffness matrices defined on these spaces with the lowest-order stiffness matrices defined on refined meshes, constructed using a Gauss-Lobatto triangular lattice. Spectral equivalence of the operators is a consequence of norm equivalences in Jacobi-weighted $L^2$ norms, which are established by proving stability of the Jacobi-Gauss-Lobatto interpolation operator in shifted norms. The low-order-refined preconditioners can also be used to precondition the standard piecewise polynomial finite element spaces using a fictitious space approach. The low-order-refined system can in turn be preconditioned effectively using algebraic multigrid methods. The analytical estimates are confirmed by numerical results on a variety of high-order problems, including on mixed meshes and surface meshes. |
| title | The high-order finite element Duffy de Rham complex and low-order-refined preconditioning |
| topic | Numerical Analysis 65N30, 65F08, 65N55 (Primary) 33C45 (Secondary) |
| url | https://arxiv.org/abs/2604.00148 |