CMPS Theory Manual › Chapter 18

18 Agglomeration Multigrid and FMG

Multigrid accelerates convergence by treating error components on spatial scales appropriate to a hierarchy of control volumes. CMPS uses worker-owned agglomerated coarse grids. The active coarse solver rediscretizes the governing equations on each level and reuses the coupled residual/Jacobian infrastructure rather than constructing a separate scalar pressure equation.

18.1 Hierarchy notation

Let level \(0\) denote the original fine grid and levels \(1,2,\ldots,L\) progressively coarser agglomerations. A fine cell belongs to one parent aggregate on the next coarse level. Denote state restriction by \(I_h^H\) and correction prolongation by \(I_H^h\).

The nonlinear equation on level \(\ell\) is \[\mathcal{N}_{\ell}(\mathbf{q}_{\ell})=\mathbf{f}_{\ell}, \](18.1) where \(\mathcal{N}_{\ell}\) is obtained by rediscretizing the active CMPS equations on that level’s geometry and boundary representation.

18.2 Correction multigrid

For a linearized fine-grid correction problem \[\mathbf{A}_h\mathbf{e}_h=\mathbf{r}_h,\] a correction cycle performs:

pre-smoothing on level \(h\);
residual evaluation \(\mathbf{r}_h=\mathbf{B}_h-\mathbf{A}_h\mathbf{e}_h\);
restriction to the coarse level;
approximate coarse solution;
prolongation of the coarse correction;
fine-state/correction update;
post-smoothing.

The smoother is not required to solve the level system accurately. Its purpose is primarily to damp error components that are oscillatory on the current grid.

After pre-smoothing, the fine-grid residual is restricted to the coarse level,

\[\mathbf r_H=I_h^H(\mathbf r_h)=I_h^H(\mathbf b_h-\mathbf A_h\mathbf x_h).\]

The coarse correction satisfies

\[\mathbf A_H\mathbf e_H=\mathbf r_H,\]

and the fine approximation is corrected by

\[\mathbf x_h\leftarrow\mathbf x_h+I_H^h\mathbf e_H.\]

The subsequent post-smoothing step damps high-frequency error reintroduced by prolongation.

18.3 Nonlinear FAS

For a nonlinear equation, Full Approximation Storage (FAS) transfers a full approximation, not only an error correction. If \[\mathcal{N}_h(\mathbf{q}_h)=\mathbf{f}_h,\] the coarse FAS equation can be written as \[\mathcal{N}_H(\mathbf{q}_H) =I_h^H\mathbf{f}_h+\boldsymbol{\tau}_H, \](18.2) with \[\boldsymbol{\tau}_H =\mathcal{N}_H(I_h^H\mathbf{q}_h) -I_h^H\mathcal{N}_h(\mathbf{q}_h). \](18.3) The \(\tau\) correction reconciles the coarse rediscretization with the restricted fine nonlinear operator. Mixing a correction-only transfer with a full FAS state without the corresponding equation transformation is mathematically inconsistent.

The FAS defect correction can be written explicitly as

\[\boldsymbol\tau_H=\mathcal N_H(I_h^H\mathbf q_h)-I_h^H\mathcal N_h(\mathbf q_h),\]

giving the coarse nonlinear problem

\[\mathcal N_H(\mathbf q_H)=I_h^H\mathbf f_h+\boldsymbol\tau_H.\]

After the coarse nonlinear solve, the correction transferred to the fine level is the difference between the coarse solution and restricted fine approximation,

\[\mathbf q_h\leftarrow\mathbf q_h+I_H^h\left(\mathbf q_H-I_h^H\mathbf q_h\right).\]

18.4 State restriction

For flow initialization and FAS, state restriction must preserve the meaning of the governing state. In compressible flow, a safe conservative restriction is based on volume-integrated conservative quantities, \[\mathbf{W}_H =\frac{1}{\Omega_H} \sum_{i\in H}\Omega_i\mathbf{W}_i, \qquad \Omega_H=\sum_{i\in H}\Omega_i, \](18.4) followed by reconstruction of primitive variables from the restricted conservative state. Independently averaging both pressure and temperature can violate total-energy conservation.

For the constant-density incompressible formulation, the stored state in Eq. (4.1) determines which components are restricted. Turbulence, species, and optional particle variables must be transferred with the same equation layout active on the target level.

For a coarse control volume assembled from fine cells \(i\in H\),

\[\Omega_H=\sum_{i\in H}\Omega_i,\]

so conservative restriction preserves the integrated state exactly:

\[\Omega_H\mathbf W_H=\sum_{i\in H}\Omega_i\mathbf W_i.\]

18.5 Correction prolongation versus full-state prolongation

Ordinary correction multigrid prolongates a correction, \[\mathbf{q}_h\leftarrow\mathbf{q}_h+I_H^h\mathbf{e}_H.\] Full-multigrid initialization instead requires a complete admissible state on the next finer level, \[\mathbf{q}_h\leftarrow \mathcal{P}_{H\rightarrow h}(\mathbf{q}_H), \](18.5) where \(\mathcal{P}\) must preserve positivity/bounds and the thermodynamic relation between primitive and conservative variables. A correction interpolator is not automatically a safe full-state interpolator.

18.6 Nested full-multigrid initialization

A full-multigrid (FMG) initializer uses nested iteration:

start from a valid initialized fine-grid state;
build/validate the hierarchy;
restrict the complete state to the coarsest usable level;
solve or apply FAS cycles on the coarsest level;
prolongate a complete state to the next finer level;
perform level/FAS work involving that level and all required coarser levels;
continue to level 0;
perform fine-grid cleanup and restore normal solver settings.

For levels \(3,2,1,0\), the nested-order backbone is \[3\rightarrow2\rightarrow1\rightarrow0,\] with FAS cycles allowed to descend again to the coarsest level from each active finer stage. Simply performing one ordinary V-cycle from the fine grid is not FMG initialization.

In nested iteration, a coarse solution supplies a full initial state to the next finer level,

\[\mathbf q_{\ell-1}^{(0)}=\mathcal P_{\ell\rightarrow\ell-1}(\mathbf q_\ell^{(*)}),\]

after which a prescribed number of FAS cycles is applied before continuing toward level zero. The work model is therefore a sum over levels rather than repeated fine-grid initialization,

\[W_{FMG}\approx\sum_{\ell=0}^{L}\nu_\ell W_\ell,\qquad W_\ell\propto N_\ell.\]

18.7 Level stopping criteria

An initialization level should stop because useful convergence has been achieved or because a defined work limit has been reached. A relative criterion is \[\frac{R_{\ell,\mathrm{current}}} {R_{\ell,\mathrm{reference}}} \le\eta_{\ell}, \](18.6) combined with a maximum cycle/iteration count. The initialization target need not be the same as the final production convergence criterion; the objective is a physically admissible starting field that substantially reduces fine-grid startup cost.

18.8 Coarse-level physics and turbulence

Rediscretization means that active boundary conditions, fluxes, sources, equation count, and thermodynamic relations have to remain meaningful on the coarse geometry. Turbulence is especially sensitive because wall distance, normals, \(k\), \(\omega\), turbulent viscosity, and wall-function quantities must refer to the current level consistently.

A robust FMG initializer may deliberately freeze or stage selected physics if a coarse representation is not valid. Such staging must be explicit and followed by a fine-grid rebuild of turbulence, wall, boundary, and property data before returning control to the normal solver.

18.9 Parallel consistency

Every worker owns its local hierarchy. A multigrid cycle must keep workers synchronized through restriction/prolongation interface exchanges and convergence reductions. If one worker detects an invalid state, the failure has to be reduced collectively before any worker destroys a level or returns to the main event loop.

The fine-grid partition itself is not replaced by FMG. Successful completion must leave level 0 active with valid primitive/conservative consistency and ready linear-solver objects.

18.10 Current implementation status

The current active multigrid implementation is under the agglomeration-multigrid solver path and receives the complete fine-grid solver options on coarse levels. Both compressible and constant-density incompressible equation layouts are connected to that dispatch. The August 2026 source review classified agglomeration multigrid as structurally connected but not runtime-validated in the audited environment. Robust FMG/FAS initialization is also present with restrictions on some physics combinations.

This chapter therefore documents the algorithmic requirements and the traced architecture, but it does not infer acceleration factors or convergence guarantees that have not been established by controlled solver benchmarks.

18.11 Current hierarchy construction and cycle controls

The 13 August 2026 source exposes three agglomeration formulations: legacy hybrid compatibility, implicit linear correction, and nonlinear FAS. Available cycle types include FMG nested iteration, V-cycle, W-cycle and F-cycle. Prolongation can use parent-cell injection, inverse-distance interpolation, or localized corrected interpolation.

Hierarchy construction supports alternative base agglomerators and an option that evaluates both candidates and retains the better admissible finite-volume mapping. Optional creation-time refinements include topology-preserving local refinement, bounded patch refinement, and a hybrid directional/patch refinement. Coarse-grid quality policy is separated from raw hierarchy creation and can warn only, automatically protect/repair, or repair and reject unresolved levels.

These geometric/agglomeration controls are distinct from the algebraic multigrid solver described in Chapter 9. The current VOF solve-readiness gate disables this agglomeration hierarchy for VOF cases.