Compare free-slip methods on Kramer annulus and spherical benchmarks

[gthyagi]

Summary

Local macOS comparison of free-slip boundary-condition implementations on the Kramer annulus and spherical-shell benchmarks.

This is Mac-only for now. The same comparison should be repeated on Gadi before drawing final conclusions about scalability or production robustness.

Benchmark drivers:

• Annulus Kramer benchmark: https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/benchmarks/annulus/ex_stokes_kramer.py
• Spherical Kramer benchmark: https://github.com/gthyagi/UW3_Annulus_Spherical_Benchmarks/blob/main/benchmarks/spherical/ex_stokes_kramer.py

Benchmark paper:

• Kramer, Davies, and Wilson (2021): https://doi.org/10.5194/gmd-14-1899-2021

Setup

Platform: local macOS workstation
MPI ranks: 8
Element pair: P2P1
Geometry: current UW3 benchmark scripts use linear mesh geometry

Annulus runs:

cellsize = 1/64
n = 2
case1 = free-slip, delta-function forcing
case2 = free-slip, smooth forcing, k = 2
methods = penalty, nitsche, rotated, constrained

Spherical runs completed so far:

cellsize = 1/8
l = 2, m = 1, k = 3
case1 = free-slip, delta-function forcing
case2 = free-slip, smooth forcing
methods completed = case1 penalty; case1/case2 nitsche, rotated, constrained

The spherical case2 penalty run was stopped before completing the solve because of very long runtime; its table row is included with unavailable metrics marked as --.

Benchmark-Paper Reference

Because the current UW3 benchmark runs use linear mesh geometry, the issue includes only the linear-geometry paper reference as a direct comparison target.

The most relevant annulus smooth reference is Kramer et al. Fig. 5, which reports the linear-geometry smooth free-slip cylindrical case. Around the corresponding 1/64 level for n=2, the approximate figure-read values are:

Case	Source	Approx. E_L2(v)	Approx. E_L2(p)	Notes
Annulus case1, delta free-slip	Fig. 3	~2.2e-3	~1.1e-1	h=1/64, n=2; reference scale only.
Annulus case2, smooth free-slip	Fig. 5	~8.0e-5	~1.1e-4	h=1/64, n=2; linear-geometry reference.
Spherical case1, delta free-slip	Fig. 4	~1.0e-1	~3.7e-1	h=1/8, l=2, m=1; reference scale only.

These values are included only as approximate benchmark-reference scales, not as strict pointwise pass/fail targets, because the geometry/discretisation details are not identical to the current UW3 linear-geometry setup.

Annulus Results

cellsize=1/64, P2P1, n=2, 8 ranks. lower leak and upper leak are absolute boundary normal-velocity errors.

Case	Method	Time (s)	E_L2(v)	E_L2(p)	lower leak	upper leak
case1	penalty	136.60	7.468e-02	1.333e-01	1.345e-07	1.185e-07
case1	nitsche	42.89	2.337e-03	1.121e-01	3.349e-07	2.556e-07
case1	rotated	25.37	2.337e-03	1.121e-01	6.680e-09	2.507e-09
case1	constrained	192.70	2.337e-03	1.121e-01	6.707e-09	4.360e-09
case2	penalty	576.40	3.077e-02	3.067e-02	5.807e-09	5.051e-09
case2	nitsche	42.67	4.307e-05	7.345e-05	2.270e-07	1.182e-07
case2	rotated	26.62	4.181e-05	7.290e-05	4.914e-09	1.327e-09
case2	constrained	259.10	4.181e-05	7.290e-05	4.970e-09	2.130e-09

Spherical Results

cellsize=1/8, P2P1, l=2, m=1, k=3, 8 ranks. lower leak and upper leak are absolute boundary normal-velocity errors.

Case	Method	Time (s)	E_L2(v)	E_L2(p)	lower leak	upper leak
case1	penalty	26640.00	6.778e-01	4.402e-01	3.409e-07	5.108e-07
case1	nitsche	90.31	1.043e-01	3.698e-01	4.001e-05	4.006e-04
case1	rotated	98.55	1.033e-01	3.692e-01	1.562e-06	4.261e-06
case1	constrained	4401.00	1.034e-01	3.692e-01	1.280e-06	2.025e-06
case2	penalty	--	--	--	--	--
case2	nitsche	71.89	2.664e-03	1.101e-02	9.005e-06	3.700e-05
case2	rotated	86.67	3.668e-03	1.104e-02	9.762e-07	3.296e-07
case2	constrained	3770.00	3.761e-03	1.140e-02	8.089e-07	2.913e-07

Main Observations

• Annulus smooth case: Nitsche, rotated, and constrained give similar volume errors and are consistent with the linear-geometry reference scale from Kramer et al. Fig. 5.
• Rotated and constrained enforce the normal-velocity condition much more strongly than Nitsche in the boundary diagnostic.
• Penalty can enforce small normal velocity but gives much worse volume errors and substantially higher cost, especially in annulus case2.
• Solver metadata: all listed standard SNES-path runs converged with PETSc reason 5 / 2; rotated uses a standalone rotated KSP path, so SNES=0 is expected and KSP reason=2, KSP iterations=1 are the relevant diagnostics.

Follow-Up

• Repeat the same comparison on Gadi.
• Revisit the spherical case2 penalty run under a controlled solver setup.
• Decide whether penalty free slip should be documented as a diagnostic/workaround rather than a reference benchmark method for curved shell benchmarks.