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A Two-Level Preconditioner Based on Dominant Components for Time-Dependent Multiscale High-Contrast Problem
Yating Wang, Yibao Li, Wing Tat Leung
TL;DR
High-contrast multiscale transient systems are difficult to solve because standard coarse spaces may miss high-conductivity features and convergence depends on contrast and fine-scale resolution. The paper develops a two-level overlapping preconditioner using an NLMC decomposition, showing that the high-permeability component can provide the global correction for sufficiently small time steps while retaining robustness and reducing coarse-problem cost.
Problem
High-contrast multiscale systems are challenging because standard coarse spaces may not capture high-conductivity channels, causing preconditioned condition numbers to scale with contrast.
Method
The paper constructs a two-level overlapping additive preconditioner whose multiscale coarse space decomposes NLMC basis functions into high-permeability and low-permeability components.
Results
The complete NLMC space preconditions the stiffness operator, while its high-permeability component captures contrast-dependent global modes and is sufficient for the transient global correction when ∆t ≲H2.
Takeaways & Limitations
Using only the high-permeability component reduces global coarse-problem dimension and computational cost while preserving robust convergence when the high-permeability volume is low.
Takeaways & Limitations
The iterative construction of the multiscale space is analyzed under an assumption that the relevant spaces have equal dimension and contraction parameter εm satisfies 0 ≤εm ≤ε∗< 1.
Abstract
from arXiv · showhide
In this work, we develop a two-level overlapping preconditioner for time-dependent problems in high-contrast multiscale media. We present a coarse-space construction based on multiscale methods, with emphasis on the relaxed nonlocal multicontinuum (NLMC) method. The NLMC space can be separated into components representing the high-permeability regions and the low-permeability background. We show that the complete NLMC space effectively preconditions the heterogeneous stiffness operator, whereas the high-permeability component alone captures the contrast-dependent global modes. For suitable small time-step sizes, the mass matrix controls the low-permeability contribution and only the high-permeability component in NLMC space is required for the global coarse correction in the two-level preconditioner. For general time-step sizes, performance can be maintained by using the full NLMC space or augmenting the high-permeability component with a standard multiscale space. The proposed coarse-space construction lowers the computational cost while preserving robustness with respect to coefficient contrast and fine-scale resolution. To further improve efficiency, the multiscale basis functions can be constructed by iteratively solving the relaxed energy-minimizing formulation. We demonstrate the robustness, efficiency and scalability of the proposed method through several numerical experiments.
1. Introduction.
The paper develops a two-level overlapping preconditioner for time-dependent high-contrast multiscale problems using an NLMC-based coarse space. Its reduced high-permeability component can lower coarse-solver cost while retaining robust convergence under suitable time-step conditions.
- Motivation: Time-dependent high-contrast multiscale systems require coarse spaces that address unresolved heterogeneity and coefficient contrast without excessive computational cost.Existing coarse-space constructions and analyses mainly concern stationary elliptic or steady-flow operators.
- Method: The proposed two-level method combines overlapping local corrections for fine-scale behavior with a multiscale global coarse correction.The mass matrix in the transient operator permits a smaller coarse space than the stiffness matrix alone.
- Coarse-space construction: The NLMC space separates high-permeability channels or fractures from the low-permeability background, with the complete space preconditioning the heterogeneous stiffness operator.The high-permeability component captures the contrast-dependent global modes.
- Time-step regimes: When ∆t ≲H2, the mass matrix controls the low-permeability contribution, so the high-permeability component alone suffices for the global coarse correction.For general time steps, the full NLMC space or an augmented high-permeability space is used.
- Efficiency: Iteratively solving the relaxed energy-minimizing formulation can improve the efficiency of constructing the multiscale basis functions.The paper presents numerical examples to demonstrate the proposed preconditioning methods.
2. Preliminary.
The preliminary formulation discretizes a high-contrast diffusion problem implicitly in time and motivates a two-level overlapping Schwarz preconditioner. The method targets contrast-independent convergence with a minimal-dimensional multiscale coarse space.
- Problem formulation: The model uses a high-contrast permeability field κ on a domain Ω with zero Dirichlet boundary and prescribed initial condition.The coefficient satisfies 0 < κmin ≤κ(x) ≤κmax.
- Fully discretized scheme: Implicit Euler time discretization produces the transient system (∆tA + M)U n+1 = MU n + F.Here A and M are the fine-grid stiffness and mass matrices, respectively.
- Motivation: The stiffness system is difficult because its condition number scales with the contrast ratio η and fine-mesh factor h−2.Preconditioning is therefore used to improve spectral properties and iterative convergence.
- Coarse-space objective: The proposed construction seeks a minimal-dimensional coarse space whose preconditioned condition number is independent of coefficient contrast.It uses an NLMC/CEM-based approach and a dominant low-dimensional component to reduce computational complexity.
- Two-level preconditioner: The two-level overlapping Schwarz method decomposes the space into overlapping local subdomains and a global coarse space with interpolation operators.Local solvers address localized errors while the coarse solver communicates globally.
3. The construction of coarse space.
The coarse space is built from relaxed NLMC energy-minimizing basis functions that distinguish high- and low-permeability continua. Iterative construction and dominant-component analysis support efficient, contrast- and mesh-robust preconditioning.
- Relaxed NLMC Space: Relaxed NLMC basis functions are obtained from an unconstrained energy-minimization problem with approximate orthogonality to local auxiliary modes.Relaxation permits computation on smaller oversampled regions while retaining approximation power.
- Preconditioner analysis: The complete NLMC space provides a coarse space for preconditioning the heterogeneous stiffness operator, with local solvers assembled from stiffness and projected s-norm matrices.The resulting condition-number estimate is independent of contrast, fine-mesh size h, and coarse-mesh size H.
- Preconditioner analysis: A stable decomposition with uniformly bounded overlap yields constants independent of contrast, h, and H for the additive Schwarz analysis.The proof combines stable local interpolation with bounded overlap.
- Iterative construction of VH: The relaxed NLMC basis can be constructed iteratively using PCG, provided the computed and exact coarse spaces have equal dimension and a controlled projection error.The required iteration count is of order log(η) under the stated exponential convergence property.
4. Fast scale preconditioner.
The proposed two-level preconditioner uses reduced coarse spaces tailored to the time-step regime, while retaining contrast-independent condition-number bounds under stated assumptions.
- Coarse-space choices: For Δt ∼ O(H^2), the coarse space can consist solely of the high-permeability component VH,1.This choice is stated as Case 1 and is sufficient for handling the high-permeability part with small time steps.
- Coarse-space choices: For general time-step sizes, the coarse space is enriched as VH,1 + WH, where WH may be a standard coarse finite-element or multiscale space.The enrichment permits larger time steps and can reduce computational cost because WH is easy to obtain.
- Preconditioner construction: The two-level additive Schwarz preconditioner combines coarse and overlapping local corrections through the coarse and local operators.The coarse and local stiffness and mass matrices enter the preconditioner through the operators associated with V0 and Vi.
- Condition-number analysis: Under the Case 1 time-step condition ctH^2 ≤ Δt ≤ CtH^2, the proposed preconditioner admits a condition-number estimate with constants independent of h, H, and coefficient contrast.The proof uses stable decomposition, bounded overlap, and control of the coarse component.
- Iterative coarse-space construction: The computed coarse space remains robust when the iterative basis construction error satisfies ε_m ≤ ε* < 1.The resulting bound uses C(ε*) independent of h, H, and coefficient contrast, and remains uniform for the stated error threshold.
- Enriched coarse spaces: Theorem 7 extends the analysis to V0 = VH,1 + WH under interpolation properties, with standard finite-element or MsFEM spaces possible for WH.Its condition-number bound follows from stable decomposition and reverse continuity based on bounded overlap.
5. Numerical Experiments.
Numerical experiments show that NLMC-based coarse spaces provide contrast-robust two-level PCG convergence with reduced coarse dimensions, while iterative basis construction preserves solver performance. The high-permeability component alone is especially effective for small time steps and extends to three-dimensional tests.
- 5.1. A time independent permeability field.: The high-permeability NLMC component alone reaches approximately 30 iterations, comparable to GMsFEM but with a much smaller dimension.This confirms its effectiveness for preconditioning high-permeability components of the stiffness matrix.
- 5.1. A time independent permeability field.: Within 7 iterations, iteratively constructed NLMC bases achieve almost the same energy-error history as bases computed by a direct solver.This improves basis-construction efficiency while maintaining two-level preconditioner iteration counts.
- 5.2. A time dependent permeability field.: For the time-dependent example, VH,1 + VH,2, VH,1 + Vms, and Vgms provide contrast-independent convergence, with VH,1 + VH,2 requiring fewer iterations than Vgms.Their coarse dimensions are 158 and 242 for VH,1 + VH,2 and Vgms, respectively, in the reported case.
- 5.2. A time dependent permeability field.: With a smaller time step, VH,1 alone yields contrast-independent convergence using a coarse-space dimension of only 57, although it requires slightly more iterations.VH,1 and Vgms are more robust than Vms and standard polynomial spaces in this case.
- 5.3. Three dimensional case.: In three dimensions, full NLMC requires 32, 27, 19, and 13 iterations for Cr = 4, 6, 8, 10, while VH,1 requires 45, 41, 30, and 21.MsFEM and gamg iteration counts grow with contrast, reaching 216 and 121 iterations, respectively, and both fail to converge at cr = 10.
6. Conlusion.
The conclusion presents a two-level preconditioner whose coarse space exploits high-permeability components, with robustness retained across time-step regimes. Numerical experiments evaluate solver iteration counts under fixed fine-mesh and fixed h/H settings.
- Numerical evaluation: Iteration counts are reported for two-level solvers with fixed fine mesh h = 1/60 and cr = 6, and with fixed h/H = 5 and cr = 4.These settings correspond to Tables 7 and 8, respectively.
- Coarse-space construction: The complete NLMC space effectively preconditions high-contrast problems, while its high-permeability component captures the contrast-dependent global modes.For suitable small time steps, the mass matrix controls the low-permeability stiffness contribution, allowing a substantially smaller coarse problem.
- Coarse-space construction: For general time-step sizes, robustness is retained by using the full NLMC space or augmenting its high-permeability component with a standard multiscale space.
- Efficient basis construction: The relaxed energy-minimizing formulation constructs NLMC basis functions more efficiently and can be solved iteratively.