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arXiv · 2603.27714

Releasing the pressure: High-order surface flow discretizations via discrete Helmholtz-Hodge decompositions

Tim Brüers·Christoph Lehrenfeld·Tim van Beeck·Max Wardetzky
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Abstract

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.

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Tim Brüers, Christoph Lehrenfeld, Tim van Beeck, Max Wardetzky. 2026-03-29. Releasing the pressure: High-order surface flow discretizations via discrete Helmholtz-Hodge decompositions. https://arxiv.org/abs/2603.27714

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