Nesting and domain-decomposition machinery (optional capstone)

[awickert]

Context

With inhomogeneous BCs as transmission data (geometric couplet = Dirichlet, mechanical couplet = Neumann) and the FFT preconditioner (#49), gFlex can support nested/downscaled solves and, optionally, parallel domain decomposition. The foundation term's exponential localization (decay length α) makes Schwarz-type iteration converge fast, and a one-way coarse→fine downscale is often accurate enough without iteration. Serial nesting/coupling is the immediate payoff; parallelism is one further use — not the point of this item.

Scope (gated — build only if nesting/scaling demand materializes)

• One-way downscale: solve coarse, impose coarse (w, slope) on a fine subdomain boundary via the Issue-4 boundary object (Interface, validation, and back-compat presets for boundary conditions #52).
• Optional two-way / overlapping-Schwarz iteration reusing the same boundary object as transmission data, with the FFT preconditioner applied per subdomain.
• Keep serial nesting/coupling working without requiring parallel infrastructure.

Acceptance criteria

• One-way nested solve reproduces the monolithic solution within tolerance away from the refinement boundary.
• (If implemented) iterative DD converges to the monolithic solution.

Dependencies

• FFT spectral preconditioner for the iterative FD solver #49 (FFT preconditioner) — applied per subdomain
• Inhomogeneous boundary-condition machinery (RHS assembly) #51 (inhomogeneous BC machinery) — RHS assembly for transmission data
• Interface, validation, and back-compat presets for boundary conditions #52 (BC interface) — boundary-data object used as the coupling interface

Notes

Optional stretch goal. Nothing in #51 or #52 waits on this issue; it shapes them only via the "expose boundary data as a first-class object" requirement already in #52.

[awickert]

Cross-reference: #58 (Domain decomposition for large-grid FD scaling) develops overlapping-Schwarz strip decomposition for performance reasons. The strip-solver machinery there could serve as a building block for the coarse→fine transmission coupling described here, which additionally requires the inhomogeneous BC stack (#51, #52).