Tim Brüers | Searcharxiv

arXiv2026

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

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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arXiv2025

Streamfunction-vorticity formulation for incompressible viscid and inviscid flows on general surfaces

This paper presents a streamfunction-vorticity formulation for the Navier--Stokes and Euler equations on general surfaces. Notably, this includes non-simply connected surfaces, on which the harmonic components of the velocity field play a fundamental role in the dynamics. By relying only on scalar and finite-dimensional quantities, our formulation ensures that the resulting methods give exactly tangential and incompressible velocity fields, while also being pressure robust. Compared to traditional methods based on velocity-pressure formulations, where one can only guarantee these structural properties by increasing the computational costs, this is a key advantage. We rigorously validate our formulation by proving its equivalence to the well understood velocity-pressure formulation under reasonable regularity assumptions. Furthermore, we demonstrate the applicability of the approach with numerical examples.

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