Multigrid–Schwarz Preconditioning

• Multigrid–Schwarz preconditioning is a family of solvers that integrates local Schwarz corrections with multigrid coarse-grid error reduction to address both high- and low-frequency error components.
• It employs tailored smoothers and coarse space strategies for diverse discretizations such as DG, spectral, and finite element methods, ensuring robust performance for SPD, saddle-point, and coupled systems.
• The method enhances parallel efficiency and scalability by balancing localized domain decomposition with global corrections, as demonstrated in applications like incompressible flow and fluid–structure interaction.

Searching arXiv for papers on multigrid–Schwarz preconditioning and closely related Schwarz–multigrid methods. Multigrid–Schwarz preconditioning denotes a class of solvers and preconditioners that couple multigrid hierarchy construction with Schwarz domain decomposition, so that local subdomain corrections provide smoothing or subspace correction and coarse-grid components provide global error transport. In the literature represented here, the term spans several distinct but structurally related constructions: monolithic multilevel overlapping Schwarz used either directly as a Krylov preconditioner or as a geometric multigrid coarse solver for incompressible non-Newtonian Stokes flow in complex geometries (Köhler et al., 31 May 2026); hybrid two-level Schwarz methods for $hp$-DG elliptic discretizations in which local Schwarz corrections are combined multiplicatively with a coarse space (Dolejsi et al., 10 Feb 2025); multiplicative Schwarz smoothers for saddle-point multigrid interpreted through successive subspace optimization (Chen, 2016); weighted Schwarz smoothers embedded in spectral-element $p$-multigrid (Stiller, 2015); non-overlapping Schwarz smoothers and two-level coarse spaces for singularly perturbed DG systems (Lorca et al., 2018); and geometric multigrid methods whose level smoothers or subsmoothers are additive Schwarz or Vanka-type solvers for immersed methods, FSI systems, or adaptive meshless discretizations (Prenter et al., 2019, Calandrini et al., 2019, Aulisa et al., 2017, Ye et al., 2022). Across these settings, the unifying purpose is to combine locality, parallelism, and coarse-scale robustness in a form appropriate to the discretization, the operator class, and the coupling structure.

1. Historical and conceptual lineage

Multigrid–Schwarz preconditioning sits at the intersection of two classical paradigms. Schwarz methods decompose a domain or algebraic graph into subproblems and combine local solves additively or multiplicatively. Multigrid methods reduce high-frequency error by smoothing and low-frequency error by coarse-grid correction. The combined viewpoint appears in several distinct formulations in the cited works.

For SPD and DG elliptic problems, a two-level hybrid Schwarz preconditioner can be read explicitly as a multigrid-like composition in which non-overlapping additive Schwarz acts as a fine-level smoother and the coarse component is applied multiplicatively rather than purely additively (Dolejsi et al., 10 Feb 2025). For saddle-point systems, a V-cycle multigrid method with multiplicative Schwarz smoothing can be interpreted as a successive subspace optimization method on the constraint space, where local constrained minimizations correspond to Schwarz patch solves and the multilevel hierarchy yields uniform convergence under stable decomposition and strengthened Cauchy–Schwarz assumptions (Chen, 2016).

In geometric high-order settings, the same synthesis appears as additive or weighted Schwarz smoothers inserted into polynomial multigrid cycles. In spectral elements, the smoother is an additive Schwarz operator over overlapping element-centered subdomains, while the multigrid hierarchy is purely in polynomial degree (Stiller, 2015). In immersed finite elements and immersed isogeometric analysis, the smoother is a Schwarz operator designed specifically to neutralize cut-element-induced near-linear dependencies, and geometric multigrid removes the remaining $h$-dependence (Prenter et al., 2019).

For coupled and indefinite systems, especially fluid–structure interaction and incompressible flow, the combination becomes more strongly “monolithic.” In incompressible non-Newtonian Stokes flow, multilevel overlapping Schwarz with a generalized Dryja–Smith–Widlund coarse space is used either directly inside FGMRES or as the coarse solver inside FEATFLOW’s geometric multigrid when the geometric coarse problem becomes too large for direct factorization (Köhler et al., 31 May 2026). In monolithic ALE FSI, geometric multigrid uses level smoothers based on modified Richardson or damped Richardson iterations preconditioned by additive Schwarz or field-split Schwarz/Vanka components (Aulisa et al., 2017, Calandrini et al., 2019). In adaptive meshless GMLS FSI, geometric multigrid is paired with multiplicative overlapping Schwarz smoothers built from physics-based splitting (Ye et al., 2022).

This suggests a broad taxonomy. One branch uses Schwarz as the smoother inside a multigrid cycle. Another uses a recursive Schwarz hierarchy itself as a coarse solver for multigrid. A third branch uses two-level Schwarz as a nonrecursive multigrid analogue. A plausible implication is that “Multigrid–Schwarz preconditioning” is best understood not as a single algorithm, but as a family of level-coupled subspace correction strategies whose exact form is discretization- and operator-dependent.

2. Core operator structures

The common algebraic pattern is the decomposition of the preconditioner into local subdomain inverses and one or more coarse corrections. In additive two-level form, one representative operator is

$$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$

with $R_i$ local restrictions, $A_i$ local operators, and $(R_0,A_0)$ defining the coarse space (Köhler et al., 31 May 2026). This formula appears in both SPD and more general settings, although in saddle-point problems the blocks and restrictions act jointly on the monolithic variable vector (Köhler et al., 31 May 2026).

For $hp$-DG elliptic discretizations, the additive Schwarz smoother is

$S^{-1} = \sum_{i=0}^{N}R_i^T\,A_i^{-1}\,R_i,$

while the symmetric hybrid Schwarz preconditioner applies the coarse correction multiplicatively:

$$P^{-1}\ =\ R_0^T A_0^{-1}R_0\ +\ \big(I - R_0^T A_0^{-1} R_0 A\big)\,\Big(\sum_{i=1}^N R_i^T A_i^{-1} R_i\Big)\,\big(I - A R_0^T A_0^{-1} R_0\big).$$

This construction is additive with respect to local components and multiplicative with respect to the mesh levels (Dolejsi et al., 10 Feb 2025).

In monolithic incompressible Stokes flow, the operator acts on the saddle-point matrix

$p$0

with local Stokes-type subdomain problems and a coarse basis built by monolithic GDSW extension (Köhler et al., 31 May 2026). The coarse basis couples velocity and pressure in the interior extension,

$p$1

so the saddle-point structure is preserved rather than approximated away (Köhler et al., 31 May 2026).

In multiplicative Schwarz smoothers for constrained minimization and saddle-point multigrid, one smoothing sweep can be represented as a sequential product of local projectors,

$p$2

where each $p$3 is the $p$4-orthogonal projector onto a local constraint subspace (Chen, 2016). This formulation makes the multigrid–Schwarz link explicit: a V-cycle is a recursion of such levelwise subspace corrections plus coarse-grid transfer.

In DG and non-overlapping settings, the local operators are often exact restrictions of the global DG bilinear form, including face terms and penalties, so “non-overlapping” refers to the subdomain partition rather than to the absence of interface coupling in the operator (Antonietti et al., 2019, Lorca et al., 2018). In immersed methods and spectral elements, the smoothers are additive Schwarz operators over overlapping support-based or element-extended patches, with local inverses exploiting tensor-product structure where available (Stiller, 2015, Prenter et al., 2019).

3. Coarse spaces and interlevel transfer

The defining feature of multigrid–Schwarz methods is the way coarse correction is built and coupled to local solves. The coarse space may be geometric, agglomerated, spectral, algebraic, or monolithic.

For incompressible non-Newtonian Stokes flow, the coarse space is the monolithic generalized Dryja–Smith–Widlund space. Interface degrees of freedom are decomposed into vertices, edges, and faces, and traces are prescribed for velocity translations and pressure constants. The interior extension is obtained by solving local saddle-point problems, and the coarse operator is the Galerkin projection

$p$5

Pressure constants on all interface entities are part of the construction, and the pressure coupling for the $p$6-discontinuous space must be supplied to FROSch (Köhler et al., 31 May 2026).

For $p$7-DG, the coarse space is a DG space on an agglomerated coarse mesh, with prolongation defined as natural injection and restriction defined as its $p$8 adjoint (Dolejsi et al., 10 Feb 2025). The same coarse-space viewpoint underlies the polytopic-grid additive Schwarz method, where the coarse grid may be non-embedded with respect to the fine grid and is generated by agglomeration and edge coarsening. There the coarse-to-fine transfer is defined variationally by $p$9 projection, precisely to accommodate non-nested fine and coarse meshes (Antonietti et al., 2019).

In fully algebraic two-level Schwarz for sparse matrices, the coarse space is spectral and local. Each subdomain solves a generalized eigenproblem of the form

$h$0

and the global coarse basis is assembled as

$h$1

This is fully algebraic and requires only local blocks of $h$2 together with diagonal lumping in the overlap (Daas et al., 2022).

In geometric multigrid, interlevel transfers may be canonical injections or discretization-aware operators. In spectral-element $h$3-multigrid, prolongation is the embedded interpolation $h$4 from lower to higher polynomial order and restriction is $h$5 (Stiller, 2015). In immersed FE/IGA, prolongation is the canonical nodal or spline refinement map, restriction is its transpose, and the coarse-grid operator is Galerkin:

$h$6

The geometry is re-intersected and quadrature recomputed on each level (Prenter et al., 2019).

In adaptive GMLS FSI, transfers are explicitly geometric and meshless. Velocity prolongation uses a divergence-free GMLS basis, pressure prolongation uses local polynomial reconstruction, and restriction is parent–child averaging across the adaptive hierarchy (Ye et al., 2022). This suggests that transfer design is one of the main axes along which multigrid–Schwarz methods specialize to the discretization technology.

4. Smoothers, local solves, and domain decomposition choices

The Schwarz component of a multigrid–Schwarz method may be additive, restricted additive, multiplicative, weighted additive, hybrid, or Vanka-type, and this choice strongly influences both parallelism and robustness.

In spectral-element multigrid, the smoother is a weighted additive Schwarz method over overlapping extended element regions. The local operator is solved by fast diagonalization, and the local corrections are blended with a diagonal weight matrix $h$7 built from a smoothed-sign ramp. The smoother is

$h$8

The use of nonuniform weights rather than arithmetic averaging improves convergence substantially, especially at higher polynomial order (Stiller, 2015).

For singularly perturbed symmetric reaction–diffusion DG systems, the smoothers are non-overlapping Schwarz operators without a coarse space at each multigrid level. Both additive and multiplicative variants are defined,

$h$9

and

$$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$0

with multiplicative smoothing giving stronger reduction but nonsymmetric preconditioned operators, hence the use of GMRES (Lorca et al., 2018).

In monolithic FSI, the smoother may be a modified Richardson iteration preconditioned by restricted additive Schwarz,

$$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$1

with natural splitting between fluid and solid blocks, LU on solid subdomains, and ILU(0) on fluid subdomains (Aulisa et al., 2017). In the field-split variant for ALE FSI, the level smoother is a damped Richardson iteration preconditioned by an additive block operator aligned with physics, and within those blocks locally multiplicative additive Schwarz with Vanka blocks is used (Calandrini et al., 2019).

For adaptive GMLS FSI, the smoother is deliberately multiplicative across physics: first a fluid block Gauss–Seidel, then an additive Schwarz correction over per-solid overlapping patches with approximate Schur complements (Ye et al., 2022). For saddle-point multigrid based on constrained minimization, multiplicative Schwarz is not merely a practical smoother but the object of the convergence theory itself (Chen, 2016).

The decomposition can also be overlapping or non-overlapping depending on the discretization. Overlapping subdomains are natural for conforming or spectral-element discretizations (Stiller, 2015) and for monolithic GDSW Stokes solvers (Köhler et al., 31 May 2026). Non-overlapping subdomains are especially natural for DG because degrees of freedom are already element-local; thus non-overlapping Schwarz smoothers and local DG restrictions avoid extra interface operators while retaining strong parallelism (Lorca et al., 2018, Antonietti et al., 2019). This suggests that the overlap strategy is often determined less by abstract Schwarz theory than by the algebraic support structure of the discrete basis.

5. Theoretical properties

The theoretical analysis of multigrid–Schwarz preconditioners varies with operator class, but several recurring principles appear: stable decomposition, strengthened Cauchy–Schwarz inequalities, overlap-to-subdomain diameter ratios, and coarse-space approximation quality.

For SPD overlapping Schwarz, one representative estimate quoted for energy-minimizing coarse spaces is

$$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$2

with $$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$3 the subdomain diameter and $$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$4 the overlap width (Köhler et al., 31 May 2026). The monolithic GDSW coarse space is described as extending these scalability properties to saddle-point systems by including pressure constraints and saddle-point interior extensions that respect inf-sup structure (Köhler et al., 31 May 2026).

For the two-level hybrid Schwarz preconditioner for $$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$5-DG, the main spectral estimate is

$$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$6

under quasi-uniform meshes, constant coefficients, convex $$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$7, and uniform degrees $$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$8 and $$M^{-1}_{\text{AS}} = R_0^\top A_0^{-1} R_0 + \sum_{i=1}^{N_s} R_i^\top A_i^{-1} R_i,$$9 (Dolejsi et al., 10 Feb 2025). The additive two-level Schwarz analogue satisfies a closely related bound with $R_i$0 במקום $R_i$1 (Dolejsi et al., 10 Feb 2025). The hybrid multiplicative use of the coarse space improves convergence in practice while preserving the same general parameter dependence.

For successive subspace optimization and multiplicative Schwarz smoothers on saddle-point problems, the key assumptions are a stable decomposition

$R_i$2

and a strengthened Cauchy–Schwarz inequality with constant $R_i$3. Under these, the energy decreases with contraction factor

$R_i$4

which is uniform with respect to mesh size and does not require full PDE regularity (Chen, 2016).

For non-overlapping two-level additive Schwarz on high-order DG over polytopic grids, the condition number depends on mesh-size ratio, polynomial degrees, and coefficient contrast. For nested meshes,

$R_i$5

while for non-nested coarse and fine grids the bound acquires a stronger $R_i$6-dependence (Antonietti et al., 2019). The coefficient-jump factor disappears if the coarse agglomeration aligns with diffusion jumps (Antonietti et al., 2019).

For fully algebraic two-level Schwarz on HPD diagonally dominant matrices, the coarse-space eigenproblem yields

$R_i$7

with $R_i$8 a coloring constant, $R_i$9 the overlap multiplicity, and $A_i$0 the eigenvalue threshold (Daas et al., 2022). The paper emphasizes that this result is proved for HPD diagonally dominant matrices, whereas non-self-adjoint and indefinite cases are supported primarily by numerical evidence (Daas et al., 2022).

For multigrid V-cycles with non-overlapping Schwarz smoothers in singularly perturbed symmetric reaction–diffusion systems, convergence theory yields energy contraction constants independent of $A_i$1, $A_i$2, and the number of levels, provided enough smoothing steps are taken; numerically, one pre- and one post-smoothing step already show level-independent convergence (Lorca et al., 2018).

A common misconception is that multigrid–Schwarz theory is uniformly mature across all operator classes. The literature here indicates otherwise. The strongest proofs concern SPD or structured DG settings (Dolejsi et al., 10 Feb 2025, Lorca et al., 2018, Antonietti et al., 2019, Daas et al., 2022), and saddle-point uniformity is well developed in certain constrained-minimization formulations (Chen, 2016). By contrast, monolithic FSI and immersed or adaptive meshless systems rely more heavily on empirical robustness than on sharp general theory (Aulisa et al., 2017, Calandrini et al., 2019, Ye et al., 2022).

6. Representative applications and performance regimes

The cited works cover a wide range of applications in which multigrid–Schwarz preconditioning is used to address either coarse-grid bottlenecks, strong coupling, cut-cell ill-conditioning, or adaptive refinement.

For incompressible non-Newtonian Stokes flow in complex geometries, the approach is implemented in the FROSch library within Trilinos/ShyLU and interfaced with FEATFLOW’s geometric multigrid (Köhler et al., 31 May 2026). The governing problem is a stationary incompressible Stokes-type system with Carreau-type viscosity, discretized with $A_i$3-discontinuous elements and solved by an outer alternating Picard–Newton scheme in which each linearized system is treated by FGMRES preconditioned by monolithic multilevel Schwarz (Köhler et al., 31 May 2026). The same FROSch instance is also used as the multigrid coarse solver, but only to low accuracy, with relative residual reduction $A_i$4 sufficing (Köhler et al., 31 May 2026). On a 3D extrusion die geometry and up to $A_i$5 MPI ranks, three-level preconditioning consistently outperformed two-level once the two-level coarse problem became large, while four-level methods were faster than two-level but slightly slower than three-level in the tested regime because the three-level coarse problem was not yet large enough to justify an additional recursion (Köhler et al., 31 May 2026).

For $A_i$6-DG elliptic problems, the hybrid Schwarz preconditioner is reported to dominate the additive one from the point of view of the speed of convergence and also computational costs (Dolejsi et al., 10 Feb 2025). Numerical results show about $A_i$7–$A_i$8 fewer CG iterations than additive Schwarz in representative Laplace and heterogeneous alternator problems, with weak scalability when the number of elements per subdomain is fixed (Dolejsi et al., 10 Feb 2025).

For spectral elements, nonuniformly weighted additive Schwarz smoothers embedded in a V-cycle with only one pre-smoothing achieve average logarithmic convergence rates in the range from $A_i$9 to $(R_0,A_0)$0, corresponding to residual reductions of almost two orders of magnitude, and reduce iteration counts by a factor of $(R_0,A_0)$1 to $(R_0,A_0)$2, leading to runtime savings of about $(R_0,A_0)$3 percent (Stiller, 2015). The method remains robust with respect to mesh size and for polynomial orders up to $(R_0,A_0)$4 (Stiller, 2015).

For singularly perturbed symmetric reaction–diffusion systems discretized by IP-DG, multilevel non-overlapping Schwarz preconditioners yield mesh-independent iteration counts and are robust with respect to the perturbation parameter $(R_0,A_0)$5 and singular reaction operators (Lorca et al., 2018). In the reported regimes, typical GMRES iteration counts are about $(R_0,A_0)$6 for two-level multiplicative Schwarz, about $(R_0,A_0)$7 for two-level hybrid, and about $(R_0,A_0)$8 for the multigrid V-cycle with multiplicative Schwarz smoother (Lorca et al., 2018).

For immersed finite element methods and immersed isogeometric analysis, additive Schwarz alone removes cut-element-induced ill-conditioning but not the usual elliptic $(R_0,A_0)$9 condition-number growth; geometric multigrid with Schwarz smoothing removes both mesh-size and cut-configuration dependence, with iteration counts effectively independent of $hp$0 and the smallest cut fraction $hp$1 (Prenter et al., 2019). The method is applicable to higher-order Lagrange bases, uniform B-splines, and THB-splines (Prenter et al., 2019).

For monolithic ALE FSI systems, geometric multigrid with Schwarz-based level smoothers provides robust performance on two- and three-dimensional benchmarks. In one formulation, modified Richardson with RAS smoothing and a direct monolithic coarse solve gives average GMRES convergence factors around $hp$2–$hp$3 in 3D and around $hp$4–$hp$5 in 2D under recommended parameters (Aulisa et al., 2017). In the field-split variant, GMG with field-split Schwarz/Vanka level solvers is about $hp$6–$hp$7 faster than a pure additive Schwarz domain decomposition preconditioner in 2D and about $hp$8–$hp$9 faster in 3D, with similar GMRES iteration counts (Calandrini et al., 2019).

For adaptive GMLS discretizations of Stokes-limit FSI, the monolithic geometric multigrid preconditioner maintains nearly constant GMRES iteration counts as the total degrees of freedom increase for a fixed number of solid bodies, and the iteration count scales nearly proportionally to $S^{-1} = \sum_{i=0}^{N}R_i^T\,A_i^{-1}\,R_i,$0 when the number of solid bodies increases (Ye et al., 2022).

7. Practical design choices, limitations, and open directions

Several practical design rules recur across the literature. First, coarse spaces must reflect the operator’s slow modes rather than merely the geometry. In saddle-point systems this means pressure constraints or exact-divergence-free structures must be represented explicitly (Köhler et al., 31 May 2026, Chen, 2016). In non-self-adjoint or indefinite algebraic settings, spectral local coarse spaces selected by generalized eigenproblems can outperform standard multigrid coarse spaces (Daas et al., 2022). In immersed or adaptive meshless problems, transfer operators must encode the discretization’s special structure, such as spline support or divergence-free reconstruction (Prenter et al., 2019, Ye et al., 2022).

Second, the smoother must match the discrete coupling graph. Weighted additive Schwarz is effective for tensor-product spectral elements (Stiller, 2015). Non-overlapping local DG solvers are natural for DG systems (Lorca et al., 2018, Antonietti et al., 2019). Vanka-like blocks or physics-based field splitting are repeatedly used for coupled incompressible flow and FSI systems (Calandrini et al., 2019, Ye et al., 2022). This suggests that “better smoother” often means “better local operator model,” not simply “more overlap.”

Third, the optimal number of levels is problem-dependent. In monolithic GDSW Stokes flow, two levels suffice up to moderate subdomain counts; three levels are preferable when the two-level coarse problem dominates runtime; a fourth level becomes useful only when the three-level coarse problem itself becomes too large (Köhler et al., 31 May 2026). Additional levels introduce overhead and may slow the solve if the recursively reduced coarse problem is not yet the dominant cost (Köhler et al., 31 May 2026).

The limitations are equally clear. Performance may be sensitive to overlap size, primal constraints, and load imbalance (Köhler et al., 31 May 2026). Very high $S^{-1} = \sum_{i=0}^{N}R_i^T\,A_i^{-1}\,R_i,$1 without corresponding coarse-space enrichment can deteriorate convergence in $S^{-1} = \sum_{i=0}^{N}R_i^T\,A_i^{-1}\,R_i,$2-DG (Dolejsi et al., 10 Feb 2025). Nonconvex domains or severe anisotropy can degrade raw multigrid performance, sometimes motivating Krylov acceleration of the V-cycle (Stiller, 2015) or specialized coarse spaces (Prenter et al., 2019). For FSI and nonlinear problems, strong empirical performance outpaces general-purpose convergence theory (Aulisa et al., 2017, Calandrini et al., 2019, Dolean et al., 2016).

Prominent future directions are named explicitly in the cited work. For the multilevel monolithic GDSW Stokes setting, larger weak-scaling studies are needed to reach regimes where four-level recursion clearly dominates, and promising directions include adaptive coarse spaces, improved partitioning to mitigate imbalance, communication-avoiding variants, and GPU-aware local solvers (Köhler et al., 31 May 2026). In algebraic two-level Schwarz for sparse matrices, extension of rigorous theory beyond HPD diagonally dominant matrices to nonnormal operators remains open (Daas et al., 2022). In spectral-element and immersed settings, a more complete spectral theory for the observed smoother behavior would strengthen the currently empirical design rules (Stiller, 2015, Prenter et al., 2019).

Taken together, these results portray multigrid–Schwarz preconditioning as a highly adaptable framework rather than a single standardized method. Its most successful realizations preserve the algebraic and physical structure of the underlying discretization, use Schwarz subproblems to eliminate locally stubborn modes, and deploy coarse spaces that capture the residual global constraints left untouched by local correction.