[jmartfrut]

Benchmark-only changes for S4-B (HyperFEM's HMultigridSolver, branch s4b-hmultigrid): the cases that build a
MeshHierarchy and the solver configurations measured against CG + GAMG. No change in src/.

Branch s4b-bench on main, 3 commits (0103492, 8f1a559, 0b2b3e3). Measurements in design/solvers-s4b-status.md.

What changes

• benchmark/solvers/cases.jl:
  • hcube_hex N K [λ]: N³ Cartesian hexahedra of the unit cube refined K times.
  • hcube_tet H K [λ]: Gmsh tetrahedra of the unit cube with mesh size H, physical groups fixedup (x = 0) and topsuf
    (x = 1) tagged with their closure (edges and vertices), refined K times. The hierarchy requires the same closed
    Dirichlet boundary on every level.
  • mech_case takes the λ of the neo-Hookean law (λ = 10, ν = 0.45, is the default of mech_cube; 1.5 gives ν = 0.3).
    The mechanics are those of mech_cube: clamped at x = 0, sheared 0.1 at x = 1, 5 load steps, order 1.
  • hem_plate_c n k: the EM plate 8 × 0.8 × 40 as (7n, 2n, 19n) Cartesian hexahedra refined k times, clamped at z = 0,
    electrodes on y = 0 (φ = 0) and y = 0.8 (φ = 0.2, the field of em_plate_c) with u_y = 0 on both (closed tags inherited
    by every level); carries the hierarchy and the test-space builders of the two fields.
• benchmark/solvers/krylov.jl: configurations hmg_<krylov>[_lagN][_degK][_rngA][_lu|_mumps] — HMultigridSolver with CG or
  GMRES, reuse = N, Chebyshev degree K, range A, coarsest level by LU, MUMPS or (default) one GAMG V-cycle with the rigid
  modes of the coarse space (PETScSolver(:preonly, :gamg; nullspace = near_nullspace(coarse trial space))).
  • fshmg_<split>[_lagN][_lu] (S4-B step 5): HyperFEM's FieldSplitSolver with one h-multigrid V-cycle per block (coarse GAMG per block,
    or LU), against S2's fs_additive on the same fine mesh.
• benchmark/solvers/run_s4b.sh: modes tet, hex (ν = 0.45), tetl, hexl (ν = 0.3) and em (hem_plate_c).

Results (our measurement: Mac M5 Max, 6 Julia threads, PETSc serial, Krylov rtol 1e-10, Newton rtol 1e-8; HyperFEM ab24cd8)

Seconds per Newton iteration (CG iterations per Newton iteration), ν = 0.45, same Newton trajectory in every run:

case	dofs	cg_gamg	hmg_cg_lag3	gain
hcube_hex 4 4 (5 levels)	799k	8.83 (29.8)	1.75 (16.8)	×5.0
hcube_tet 0.15 3 (4 levels)	414k	5.70 (52.8)	2.67 (32.2)	×2.1
hcube_tet 0.12 3 (4 levels)	893k	15.97 (84.7)	2.80 (31.9)	×5.7

EM plate (u, φ), GMRES: hem_plate_c 1 3 (555k) fs_additive 10.14 (97.5), fshmg_additive_lag3 4.09 (55.8),
fshmg_schur_lag3 3.33 (49.9): ×2.5–3.0; hem_plate_c 1 2 (71k) 0.84 vs 0.71: ×1.2.

Requires HyperFEM with HMultigridSolver (branch s4b-hmultigrid; the benchmark environment develops ../HyperFEM.jl).