A Kernel-Aware Two-Grid Preconditioner for Cell-Centered Nearly Incompressible Elasticity

1. Introduction

• Local stress and rotation elimination produces a sparse symmetric positive definite system with one displacement vector per cell, while mixed fields are recovered by local back-substitution.The reduction retains local conservation and locking-free approximation.Source: §1 · p. 2
• As the bulk-to-shear ratio increases, the reduced Schur complement increasingly penalizes volume-changing displacement components relative to nearly volume-preserving ones.Identifying the discrete volumetric constraint within the reduced operator is therefore central to preconditioning.Source: §1 · p. 2
• Conventional cell-centered interpolation may introduce volumetric energy when prolongating volume-preserving coarse displacements, with the defect amplified by the bulk modulus.A constraint-preserving coarse correction complements local relaxation by treating these modes globally.Source: §1 · p. 2
• The two-subspace coarse correction combines conventional multilinear displacement functions with potential-generated functions that remain in the fine-grid volume-preserving subspace.Vertex-local balance relations define discrete volume change, and discrete curls of prolonged potentials produce the kernel-compatible transfer.Source: §1 · p. 2
• The energy splitting and kernel-compatible decompositions yield a condition-number bound independent of the Lamé ratio for each fixed grid pair.An energy comparison extends the bound to the symmetric algorithmic composition.Source: §1 · p. 3
• In two and three dimensions, iteration counts remain controlled across eight orders of magnitude in Lamé ratio on each grid, with moderate growth under refinement.Removing the potential-generated correction produces pronounced parameter dependence; parallel tests also exceed fifty million displacement unknowns.Source: §1 · p. 3

2. Model problem and cell-centered discretization

• The section sets up the model and discretization whose locally reduced cell-centered displacement operator is used by the preconditioner.Stress and rotation are eliminated locally to obtain that operator.Source: §2 · p. 3

2.1. Weakly symmetric mixed formulation.

• The model uses a weakly symmetric stress–displacement–rotation formulation for elasticity in two or three dimensions, with Lamé parameters λ and µ.The rotation is the skew part of the displacement gradient for a sufficiently smooth solution.Source: §2.1 · pp. 3–4

2.2. Discrete spaces and mixed scheme.

• The mesh is partitioned into subcells, with displacement constant on each macro-element and rotation constant on each vertex interaction region.The stress space is subcellwise constant and its basis functions are supported on individual interaction regions, localizing the compliance form.Source: §2.1 · p. 4, §2.2 · p. 5

2.3. Local elimination of stress and rotation.

• Stress and rotation are eliminated through vertex-local solves, leaving a symmetric positive definite system for cell-centered displacement under the stated rank and inf–sup conditions.Stress and rotation are recovered sequentially by local back-substitution after the global displacement solve.Source: §2.3 · p. 6

2.4. Three-dimensional local construction.

• In three dimensions, the reduced displacement matrix is assembled by summing vertex contributions whose inverses act on uniformly bounded-size matrices.The local construction covers interior and boundary vertices; the second elimination is well defined for each reference-patch type.Source: §2.4 · p. 7

3.1. Discrete volumetric operator and kernel.

• Vertex balance relations define discrete volume change, whose kernel is reached by applying the discrete curl to prolonged potentials.The discrete complex gives DhKh = 0, so transfers built this way remain in the fine-grid incompressible kernel.Source: §3.1 · p. 8

3.2. Kernel-aware coarse correction.

• The global correction combines a geometric displacement space with a kernel-compatible space; their independently factored Galerkin corrections supply the global component.Geometric interpolation approximates smooth displacement but is not generally kernel preserving.Source: §3.2 · pp. 8–10

3.3. Vertex-patch relaxation.

• Vertex-patch correction spaces use cells surrounding each fine-grid vertex to resolve coupling within the interaction regions.The patches are built from the same regions that determine assembly of Ah.Source: §3.3 · p. 10

3.4. Symmetric two-grid composition.

• The two-grid composition combines global coarse correction with forward and backward patch sweeps to reduce global and fine-scale error components.The composition is given by BTL = S + (I − STAh)BH(I − AhS).Source: §3.4 · p. 10

4.1. Parameter structure of the local Schur complements.

• The local spherical-mode energy is uniformly equivalent to its λ=0 counterpart, with a constant independent of mesh size and Lamé coefficients.The bound is r0,a(vh,vh) ≤ rλ,a(vh,vh) ≤ Cloc r0,a(vh,vh).Source: §4.1 · p. 13
• A mesh-independent stress–displacement coupling bound and a block criterion reduce three-dimensional shear stability to a uniform lower bound for the reduced λ=0 energy.The criterion assumes full row rank of the reference-patch rotation matrices.Source: §4.1 · p. 14
• The smallest generalized eigenvalue decreased from 1.423 to 1.022 under refinement, providing quantitative evidence of shear-energy control in the jump norm.The reported maximum relative eigenpair residual was 5.3 × 10^-11.Source: §4.1 · p. 15, Table 2 · p. 15
• Under Assumption 4.1, the reduced form splits exactly into a shear term and (λ/µ) times the squared volume change, with the shear remainder controlled in a fixed norm.The volume-change operator is Dh, so its kernel is the limiting constraint space.Source: §4.1 · p. 16

4.2. Exactness of the cell-centered complex.

• The discrete curl potentials exactly span the volumetric kernel, and the kernel transfers have full column rank and annihilate Dh uniformly in h and λ/µ.The exact representation is range(Kh) = ker(Dh).Source: §4.2 · pp. 17–18
• Both coarse transfers lie in the fine-grid volumetric kernel, and their coarse-degree-of-freedom maps are injective in two and three dimensions.In three dimensions, injectivity follows by telescoping fine gradients along coarse edges and using the spanning-tree gauge.Source: §4.2 · p. 18

4.3. Decompositions over the two coarse spaces and patches.

• General displacements admit a shear-stable coarse-plus-patch decomposition, while kernel displacements admit an exact patch decomposition with finite stability constant in both dimensions.Adding the coarse kernel space cannot increase the kernel decomposition constant.Source: §4.3 · pp. 19–20

4.4. The additive split operator and the hybrid method.

• The symmetric hybrid correction is spectrally equivalent to the additive split correction, with comparison constants independent of h, λ, and µ for fixed coloring and smoothing parameters.The hybrid operator is symmetric positive definite and may be used with conjugate gradients.Source: §4.4 · pp. 21–22

4.5. Parameter-uniform condition-number bound.

• For a fixed nested grid pair, Algorithm 3.1 has a condition-number bound independent of λ/µ, and its PCG energy-error contraction is controlled uniformly in the Lamé ratio.The bound depends on the coarsening ratio, patch enlargement, and finite-color constants.Source: §4.5 · pp. 23–24

5. Numerical experiments

• The experiments use exact solutions to relate solver accuracy to approximation error, and compare both coarse-space choices across mesh and material variations.The study includes two- and three-dimensional cases, material interfaces, and nonhomogeneous displacement data.Source: §5 · p. 24

5.1. A smooth nearly incompressible solution.

• On the 1282 grid at λ/µ = 108, the enriched method requires 28 applications versus 197 for displacement correction alone.For the smooth two-dimensional problem, errors decrease at first order through 5122, while the enriched PC count rises from 15 to 28.Source: §5.1 · pp. 25–26, Table 4 · p. 25

5.2. Discontinuous material coefficients.

• 35 versus 350 applications at λ/µ = 108 on the 962 grid: the two-subspace method retains its advantage with discontinuous coefficients.Interface-fitted refinement preserves first-order convergence for all four fields, including physical rotation recovered from the weighted space.Source: §5.2 · p. 26, Table 5 · p. 26, Table 6 · p. 27

5.3. A homogeneous oscillatory solution in three dimensions.

• 159 to 30 applications at λ/µ = 106 on the 643 grid when adding the kernel correction.The three-dimensional refinement sequence also retains first-order accuracy for all four fields through 50,331,648 displacement unknowns.Source: §5.3 · pp. 27–28, Table 8 · p. 28

5.4. Nonhomogeneous displacement data in three dimensions.

• 144 to 25 applications at λ/µ = 106 on the 643 grid with the kernel correction under prescribed boundary deformation.The three reported fields retain approximately first-order convergence through 2563, and enriched counts remain controlled toward the incompressible limit.Source: Table 9 · p. 28, §5.4 · p. 29, Table 10 · p. 29

• The two-grid preconditioner pairs local mixed-variable elimination with a potential-generated coarse space that preserves the discrete incompressibility constraint.Symmetric vertex-patch sweeps resolve local coupling; the energy decomposition supports parameter-uniform subspace-correction estimates.Source: §6 · p. 29