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

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.

Figures (7)

• Figure 1: Comparison of the standard and "hol(e)y" Stanford Bunny. Both surfaces depict a divergence-free vector field alongside contours of the corresponding scalar streamfunction. The left bunny is simply connected, and thus no harmonic fields are present. Consequently, the vector field aligns perfectly with the contours of the streamfunction. In contrast, the right bunny has the topology of a torus, and thus the harmonic space has dimension $b_1(M)=2$. Here, the vector field contains a non-trivial harmonic component and therefore clearly deviates from the contours of the streamfunction.
• Figure 2: We approximate a surface $M \subset \mathbb{R}^3$ by a discrete surface $M_h$. The approximation of the geometry may be linear (above left) or of higher order (below left) where the latter is typically obtained from the former by a piecewise smooth map $\Psi$.
• Figure 3: Decomposition \ref{['eq:HkDecomp']} of an $\mathbf{H}^1$ vector field into a gradient of a potential, rotation of a streamfunction, and a harmonic field (from left to right) on a torus.
• Figure 4: Visualization of the discrete decomposition \ref{['eq:BDMDecomp']} of a vector field in $\mathds{BDM}^k_0$ into a harmonic and rotation field and remaining complement (from right to left) on the "bob" geometry from crane2013robust. Computed through the procedure explained in \ref{['sec:harmonic:algo']} below.
• Figure 5: Snapshots of the surface Navier--Stokes flow on the thickened trefoil knot at times $t=0,\,100,\,200,\,400,\,800$. Each column shows the total velocity $\bm{u}_h$ (top), the rotational part $\bm{u}_{\hbox{$\mathop{\mathrm{\mathbf{rot}}}\nolimits$}} = \mathop{\mathrm{\mathbf{rot}}}\nolimits(\psi)$ (middle, arrows scaled $\times 2$), and the harmonic part $\bm{u}_{\hbox{$\mathds{H}$}}$ (bottom) . The colors indicate velocity magnitude, scaled from $0$ to $5$. A full animation is available in grodata and at https://www.youtube.com/watch?v=DqHXLNkhF0o.

Theorems & Definitions (11)

• Remark 3.1: Mixed boundary conditions
• Theorem 3.2: Decomposition of $\mathds{J}^k_{\mathds{BDM}}$ and $\mathds{BDM}^k_0$
• Remark 3.3: Hierarchy of the $\mathds{BDM}_0^k$ complex
• Remark 3.4: $H^2$-complex in the planar case
• Remark 3.5: Curved triangulations
• Remark 4.1
• Remark 5.1: Pressure reconstruction via incomplete discrete Helmholtz--Hodge decomposition
• Theorem Appendix B.1: Dimension of $\mathds{H}_{\mathds{BDM}}^k$
• proof
• Theorem Appendix B.2: Dimension of $\mathds{H}_{\mathds{SV}}^k$