Multigrid API

abstract type AbstractMultigridMode

Controls operator assembly in MultigridData construction.

• DirectMode(): dualize and discretize at every level during the walk.
• GalerkinMode(): only dualize the finest mesh; derive coarse operators via the Galerkin condition Aₖ₊₁ = Rₖ Aₖ Pₖ.

struct MeshTopology

Boundary maps of a simplicial 2-complex as plain arrays.

MultigridData(s, scheme, levels, op, steps; alg, mode)

Construct a MultigridData from a base mesh using topology-only subdivision.

Keyword arguments

• mode::AbstractMultigridMode = DirectMode(): controls operator assembly.
• alg = Circumcenter(): dualization algorithm.

Examples

md = MultigridData(s, BinarySubdivision(), 4, s -> ∇²(0, s), 3)
md = MultigridData(s, BinarySubdivision(), 4, s -> ∇²(0, s), 3; mode=GalerkinMode())

Direct mode: operators were collected coarsest-first; reverse to finest-first.

Galerkin mode: dualize the finest mesh, derive coarse operators via Rₖ Aₖ Pₖ.

Compute the coarsest-level operator before the walk begins.

No operators needed before the walk in Galerkin mode.

Dualize the current mesh and record its operator.

No per-level operators in Galerkin mode.

binary_subdivision_topo(topo::MeshTopology) -> (refined::MeshTopology, mat)

Binary subdivision returning both topology and matrix.

cubic_subdivision_topo(topo::MeshTopology) -> (refined::MeshTopology, mat)

Cubic subdivision returning both topology and matrix.

Full multigrid: start at the coarsest grid and work up, applying μ-cycles at each level (μ=1 for V, μ=2 for W).

make_restriction(p::Transpose{<:Any, <:SparseMatrixCSC})

Build restriction R = row_normalize(S) directly from the subdivision matrix S = parent(p), sharing its colptr and rowval arrays. Only allocates a new nzval vector and a temporary row-sum buffer.

Solve Ax=b with initial guess u for cycles V-cycles. alg is a Krylov.jl method (default cg).

W-cycle variant of multigrid_vcycles.

propagate_points(::BinarySubdivision, topo, coarse_points) -> fine_points

Interpolate vertex positions for binary subdivision. Original vertices are copied; midpoints are averaged from edge endpoints.

propagate_points(::CubicSubdivision, topo, coarse_points) -> fine_points

Interpolate vertex positions for cubic subdivision. Original vertices are copied; edge points at ⅓/⅔ positions and triangle centroids are computed from connectivity.

propagate_points(mat::SparseMatrixCSC, coarse_points) -> fine_points

Interpolate via column-wise SpMV on the subdivision matrix. Exported for external callers that have a matrix but no scheme type; all internal paths use the scheme-dispatched methods above.

refine(::BinarySubdivision, topo) -> MeshTopology

Binary (medial) topology refinement: split each edge at its midpoint, subdivide each triangle into 4. No subdivision matrix is constructed.

refine(::CubicSubdivision, topo) -> MeshTopology

Cubic topology refinement: two interior points per edge plus a centroid per triangle, subdividing each triangle into 9. No subdivision matrix is constructed.

repeated_subdivision_maps(k, ss, subdivider) -> Vector{PrimalGeometricMap}

Apply k subdivisions via subdivider, returning geometric maps (meshes + matrices). Used by PrimalGeometricMapSeries.

repeated_subdivisions(k, ss, scheme) -> Vector{EmbeddedDeltaSet}

Apply k subdivisions. Returns meshes only — no subdivision matrices are constructed.

subdivision(s, scheme) -> EmbeddedDeltaSet

Subdivide a mesh. Only refines topology and propagates points — no subdivision matrix is constructed.

subdivision_map(s, scheme) -> PrimalGeometricMap

Subdivide and return the geometric map (mesh + subdivision matrix).

subdivision_matrix(::BinarySubdivision, topo) -> SparseMatrixCSC

Build the binary subdivision matrix (nvcoarse × nvfine) in direct CSC format. Identity block for original vertices, ½/½ averages for midpoints.

subdivision_matrix(::CubicSubdivision, topo) -> SparseMatrixCSC

Build the cubic subdivision matrix (nvcoarse × nvfine) in direct CSC format. Identity block for original vertices, ⅓/⅔ weights for edge points, equal weights for centroids.

topo_to_mesh(::Type{S}, topo::MeshTopology, points) -> S

Reconstitute an EmbeddedDeltaSet from a MeshTopology and point data.