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

TL;DR

The paper introduces a discrete Helmholtz--Hodge decomposition for H(div)-conforming finite elements on surfaces, enabling pressure-free, divergence-free flow discretizations with exact tangentiality and pressure robustness.

Contribution

It develops a novel decomposition that simplifies incompressible flow discretization on surfaces by eliminating pressure and saddle-point structures.

Findings

The method achieves divergence-free, pressure-robust flow discretizations.

Numerical experiments on complex surfaces demonstrate the method's effectiveness.

The approach provides physical insights into harmonic velocity components.

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.