Three-Dimensional Brinkman MMS Report

The 3D manufactured case exercises tetrahedral assembly, vector derivatives, pressure stabilization, and the triangular-facet coefficient independently of the 2D implementation.

Exact divergence-free solution

On \(\Omega=(0,1)^3\), let

\[ B=x^2(1-x)^2y^2(1-y)^2z^2(1-z)^2,\qquad \phi=32B(1+x+2y+3z), \]

\[ \mathbf u_{\rm ex}= \begin{bmatrix} 3\phi_y-2\phi_z\\ \phi_z-3\phi_x\\ 2\phi_x-\phi_y \end{bmatrix},\qquad p_{\rm ex}=\sin(2\pi x)\sin(\pi y)\sin(\pi z). \]

The mixed derivatives cancel in \(\nabla\cdot\mathbf u_{\rm ex}\), and the bubble makes velocity zero on all six faces. With \(\nu=10^{-2}\) and \(\gamma=1\), voids manufactures the same strong residual as in the 2D report. The exact velocity is imposed on the complete boundary, and pressure error is compared modulo its mean.

The image is a \(z=1/2\) section through the analytic 3D functions; it is an illustration, not a sampled finite-element result.

Five-level convergence study

The meshes use \(n=(4,6,8,10,12)\) subdivisions per direction and contain \((384,1296,3072,6000,10368)\) tetrahedra. Taylor--Hood, P1/DG0 USFEM, and P1/DG1 USFEM are compared on the identical mesh sequence. The USFEM report profile uses facet_size_mode="representative", hence \(h_F=\sqrt{2}/n\), and a 24-level reference-face solve for face3d.

Method	\(r(L^2_u)\)	\(r(H^1_u)\)	\(r(L^2_p)\)	Finest \(L^2_u\)	Finest \(L^2_p\)
Taylor--Hood P2/P1	4.088	2.884	2.088	\(2.498\,10^{-3}\)	\(6.313\,10^{-3}\)
USFEM P1/DG0	1.901	1.121	1.326	\(1.968\,10^{-2}\)	\(5.894\,10^{-2}\)
USFEM P1/DG1	1.901	1.004	1.131	\(1.307\,10^{-2}\)	\(1.441\,10^{-2}\)

Each curve contains five live solves and each triangle reports a measured finest-pair slope. The two linear-velocity formulations recover approximately second-order \(L^2\) and first-order \(H^1\) behavior. Taylor--Hood is superconvergent in velocity for this symmetric polynomial case; the nominal portable expectation remains \(3/2/2\), not \(4/3/2\). The raw table is available as CSV.

The 3D face3d law comes from a numerical reference-triangle subproblem. Its observed convergence is evidence for the implemented structured-tetrahedron study, not a general stability proof for arbitrary anisotropic tetrahedra. The presentation-replication profiles additionally provide the longer \((4,6,8,10,12,16,20)\) sequence for report-value regression.

Executable MWE

examples/fem_mms/mms_3d.py recreates the five-level table, analytic section, convergence plot, and rate assertions:

pixi run python examples/fem_mms/mms_3d.py \
  --output-dir examples/outputs/fem_mms/3d

The terminal \(12^3\) Taylor--Hood and P1/DG1 direct solves are intentionally nontrivial. For fast smoke testing, shorten the sequence; do not present the resulting coarse slopes as the documented convergence study.