Polygonal macro meshes

PolygonMesh accepts conforming partitions of straight-sided simple polygons, including nonconvex cells and collinear boundary vertices. Each macrocell has its own conforming triangular local mesh. The global unknowns remain on the original polygon edges; local triangulation edges do not become macrofaces.

The shared Darcy, conservative transport, Stokes–Brinkman and MsHHO operators are available through solve_darcy_polygons, solve_transport_polygons, solve_brinkman_polygons and solve_mshho_polygons. The automated checks include an L-shaped macrocell, polynomial patches, oriented normal fluxes and MHM–MsHHO field equivalence. Hanging macro vertices must be inserted consistently in both incident cells. Cells with holes require a partition into simple polygons.

Oscillatory transport on five polygon families

The data and approximation degrees follow section 5.2.1 of the Araya et al. (2024):

\[ u=\sin(6\pi x)\sin(14\pi y),\qquad A=0.1I,\quad\beta=(1,0),\quad c=0, \]

\[ f=0.1(6^2+14^2)\pi^2u+ 6\pi\cos(6\pi x)\sin(14\pi y). \]

Dirichlet data are homogeneous on the unit square. The skeletal degree is one and local finite elements have degree three. The study constructs triangular, square, rhombic, L/square and clipped hexagonal partitions at five resolutions. local_triangulation="boundary" gives distinct adjacent triangles to distinct macrofaces, as required by the construction in assumption (A1). An interior centroid fan is used when it lies in the polygon kernel; otherwise, each ear triangle has its own centroid fan. There is no extra local refinement in this P3/P1 campaign, consistent with (A2). These explicit meshes and refinement choices define this reproduction of the PDE and spaces; the publication does not supply their complete local connectivity, so pointwise equality with its historical mesh is not asserted.

The coarsest meshes underresolve the seven vertical oscillations. Error decreases as the macro partition resolves them. Comparisons at the same partition parameter do not imply the same number of cells or unknowns: the machine-readable record also contains those counts. This is a convergence study, not a ranking of polygon families at equal computational cost.

The analytical and numerical fields share a color scale. The signed error has a separate scale, and every panel shows the actual macro boundaries. Display sampling evaluates the cubic local functions independently on each fine triangle. All error norms use volume quadrature rather than image samples.

Run pixi run --locked -e notebooks verify-polygons. The records are stored in examples/results/polygons.json. These research runs are separate from the small polynomial and geometry checks used in CI.

References

• Rodolfo Araya, Fabrice Jaillet, Diego Paredes, and Frédéric Valentin (2024). Generalizing the multiscale hybrid-mixed method for reactive-advective-diffusive equations, Computer Methods in Applied Mechanics and Engineering 428, 117089. DOI: 10.1016/j.cma.2024.117089.