feat(boundaries): Robin BC (a·u + b·∂u/∂n = c)

[kylebeggs]

What's missing

AbstractBC is exactly Periodic, Dirichlet, Neumann, Interface. _require_face_bc (src/boundaries.jl:82) errors loudly for anything else, so a Robin condition is a genuine extension rather than a config option — it needs _bc_sign, _offset_ghost! and a _require_face_bc method, plus the matching fold_bc! row and the operator_diagonal correction.

Why (downstream)

Robin is the native boundary condition for trans-wall exchange in the microvascular model, in two places:

1. Perivascular wall coupling. Flux across the capillary wall is g·(C_blood − C_tissue) — a conductance times a concentration difference. That is a Robin condition on the tissue side, currently hand-assembled as a separate coupling operator with its own implicit/explicit split.
2. Proton leak regularization. The transported proton field H is no-flux on every boundary, which makes its Laplacian pure-Neumann and therefore singular. A small Robin leak gH is what regularizes it. Right now that regularization is a bespoke term rather than a boundary condition, so it is invisible to operator_diagonal and to multigrid.

Both are the same mathematical object, and both would be one BC type if MFO had it.

Sketch

Robin(a, b, c)      # a·u + b·∂u/∂n = c ; Dirichlet is b=0, Neumann is a=0

Worth deciding up front whether a, b, c may be per-face Fields or must be scalars. The downstream conductance g is genuinely per-face (it varies with local vessel geometry), so scalar-only would leave the main use case unserved — but per-face coefficients interact with the affine boundary_rhs lift, so it's a real design call, not a detail.

Payoff beyond the downstream package

operator_diagonal gaining a Robin case is what lets MultigridPreconditioner precondition wall-coupled systems. That matters: the downstream steady solve currently reaches for ILU(0) on an assembled sparse W precisely because unpreconditioned matrix-free GMRES spent ~750 iterations per linear solve. The system is diffusion-dominated — ideal MG territory — so a Robin-aware diagonal is the step that makes MFO's native preconditioner applicable and lets the sparse-matrix detour go away.

---

_{🤖 Beep — Claude wrote this one; Kyle is not yet talking to himself in his own issue tracker.}

[kylebeggs]

Prior art for this issue, surfaced from a cardiac GPU paper

Reading Kaboudian, Cherry & Fenton, Computing in Cardiology 50 (2023) turned up the reference this issue is missing. Their zero-flux boundary treatment cites:

Chai, Luo, Wang, Zheng & Fan, "Imposing mixed Dirichlet–Neumann–Robin boundary conditions on irregular domains in a level set / ghost fluid based finite difference framework," Computers & Fluids 214:104772 (2021). DOI

That is Robin worked out in a finite-difference framework, unified with Dirichlet and Neumann as one parameterized family — which is exactly the Robin(a, b, c) shape sketched above, where b=0 recovers Dirichlet and a=0 recovers Neumann. Useful for the two things this issue leaves open:

1. How the ghost value is constructed. Our _offset_ghost! (src/boundaries.jl:262-275) gives Dirichlet 2·value and Neumann layer k a (2k-1)·Δ·flux offset. Robin's ghost is a combination whose weight depends on a, b, and Δ — and getting that weight is what determines both the _bc_sign analogue and the operator_diagonal correction. That derivation is in the paper.
2. Whether a, b, c may be per-face fields. The paper's framework carries spatially varying coefficients on the boundary as a matter of course, since the boundary is a level set rather than a face of a box. That is evidence the per-face-Field answer — which the body notes is what the downstream conductance g actually needs — is tractable rather than exotic.

Wider relevance: this is the #62 ∩ #69 intersection

The reason the citation lands in a cardiac voxelization paper rather than a boundary-conditions paper is that Robin-on-irregular-domains is the same problem as #69's option 4 (level-set / embedded boundary). If #69 ever evaluates option 4 seriously, this paper serves both issues, and a Robin BC on a curved tissue surface is the natural joint deliverable.

That is a reason to keep the two loosely coupled rather than merged: Robin on the six box faces — which is all the microvascular model needs, per the body above — is a self-contained, much smaller job that does not need any of #69's machinery. Doing box-face Robin first is still right; this just records where the harder version's prior art lives.

---

_{🤖 Beep — Claude went looking for a heart and came back with a citation.}