Releasing the pressure: High-order surface flow discretizations via discrete Helmholtz-Hodge decompositions
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arXiv
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| Format: | Preprint |
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2026
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| author | Brüers, Tim Lehrenfeld, Christoph van Beeck, Tim Wardetzky, Max |
| author_facet | Brüers, Tim Lehrenfeld, Christoph van Beeck, Tim Wardetzky, Max |
| contents | We present a discrete Helmholtz--Hodge decomposition for H(div)-conforming Brezzi--Douglas--Marini (BDM) finite elements on triangulated surfaces of arbitrary topology. The divergence-free BDM subspace is split L2-orthogonally into rotated gradients of a continuous streamfunction space and a finite-dimensional space of discrete harmonic fields whose dimension equals the first Betti number of the surface. Consequently, any incompressible flow discretized on this subspace can be reformulated with a scalar streamfunction and finitely many harmonic coefficients as the only unknowns. This eliminates the pressure and the saddle-point structure while ensuring exact tangentiality, pointwise divergence-freeness, and pressure-robustness. We present a randomized algorithm for constructing the harmonic basis and discuss implementation aspects including hybridization, efficient treatment of the harmonic unknowns, and pressure reconstruction. Numerical experiments for unsteady surface Navier--Stokes equations on a trefoil knot and a multiply-connected sculpture surface demonstrate the method and illustrate the physical role of the harmonic velocity component. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27714 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Releasing the pressure: High-order surface flow discretizations via discrete Helmholtz-Hodge decompositions Brüers, Tim Lehrenfeld, Christoph van Beeck, Tim Wardetzky, Max Numerical Analysis Differential Geometry 65N30, 76D05, 58A14, 76D07, 76M10, 35Q30, 58J10 We present a discrete Helmholtz--Hodge decomposition for H(div)-conforming Brezzi--Douglas--Marini (BDM) finite elements on triangulated surfaces of arbitrary topology. The divergence-free BDM subspace is split L2-orthogonally into rotated gradients of a continuous streamfunction space and a finite-dimensional space of discrete harmonic fields whose dimension equals the first Betti number of the surface. Consequently, any incompressible flow discretized on this subspace can be reformulated with a scalar streamfunction and finitely many harmonic coefficients as the only unknowns. This eliminates the pressure and the saddle-point structure while ensuring exact tangentiality, pointwise divergence-freeness, and pressure-robustness. We present a randomized algorithm for constructing the harmonic basis and discuss implementation aspects including hybridization, efficient treatment of the harmonic unknowns, and pressure reconstruction. Numerical experiments for unsteady surface Navier--Stokes equations on a trefoil knot and a multiply-connected sculpture surface demonstrate the method and illustrate the physical role of the harmonic velocity component. |
| title | Releasing the pressure: High-order surface flow discretizations via discrete Helmholtz-Hodge decompositions |
| topic | Numerical Analysis Differential Geometry 65N30, 76D05, 58A14, 76D07, 76M10, 35Q30, 58J10 |
| url | https://arxiv.org/abs/2603.27714 |