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Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms
Y. Efendiev, J. Galvis, R. Lazarov, J. Willems
TL;DR
The paper addresses robust preconditioning for symmetric positive definite problems with large spatial scales and high-contrast coefficients, especially where existing scalar elliptic constructions do not extend easily to vector and mixed formulations. It develops an abstract eigenvalue-based stable-decomposition framework for general symmetric bilinear forms, applies it across elliptic and flow problems, and reports robust numerical behavior, while noting unresolved estimates and a practical interpolation gap.
Problem
High-contrast parameters and large problem sizes produce poorly conditioned systems, while existing coarse-space constructions are not readily applicable to vector problems and more complicated discretizations.
Method
The paper constructs coarse spaces by selecting local generalized eigenfunctions within an abstract stable-decomposition framework for general symmetric bilinear forms.
Results
The numerical experiments show robust preconditioners for highly variable elliptic problems, contrast-independent iteration counts for Brinkman equations, and small coarse spaces in complex geometries.
Takeaways & Limitations
The framework applies to scalar elliptic Galerkin and mixed formulations, Stokes equations, and Brinkman equations, with stable decompositions robust to contrast and mesh parameters in the analyzed settings.
Takeaways & Limitations
Rigorous L∞ estimates needed for some dimension-reduction verification remain an open problem, and the interpolation option used computationally is not immediately covered by the abstract setting.
Abstract
from arXiv · showhide
An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into "local" subspaces and a global "coarse" space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function formulation of its mixed form, Stokes' and Brinkman's equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Efendiev and Galvis, Multiscale Model. Simul., 8 (2010), pp. 1461--1483, developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman's problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.
1. Introduction
The paper develops a general framework for robust domain decomposition preconditioners targeting large problem sizes and high-contrast physical parameters. It extends eigenvalue-based coarse spaces to general symmetric bilinear forms and demonstrates applications and numerical robustness across several flow and elliptic formulations.
- Motivation: High-contrast coefficients and fine spatial scales produce large discrete condition numbers, complicating iterative solution of symmetric positive definite problems.The motivating examples include thermal conductivity and porous-media permeability variations over several orders of magnitude.
- Related work: Existing eigenvalue-based coarse-space methods handle heterogeneous scalar elliptic problems but are not readily applicable to vector problems and more complicated discretizations.The paper identifies the need for a framework that avoids constructing a separate problem-specific eigenvalue formulation for each differential equation.
- Contribution: The proposed framework constructs stable decompositions into local and coarse subspaces for general symmetric bilinear forms, with explicit bounds for the stability constant.The coarse spaces are derived from generalized local eigenvalue problems and are designed for robust additive Schwarz preconditioning.
- Applications: The framework covers Darcy, Stokes, and Brinkman formulations, including Darcy–Stokes behavior in Brinkman flow and mixed formulations that preserve local mass conservation.Brinkman’s equations combine high-flow Stokes regions with slow-flow Darcy regions, making them inherently high-contrast.
- Numerical results: Numerical experiments report robust preconditioners for highly variable elliptic problems, contrast-independent iteration counts for Brinkman discretizations, and small coarse spaces for complex geometries.Multiscale initial basis functions also enable substantial coarse-space dimension reduction for many small isolated high-conductivity inclusions.
2. Problem Setting and Notation
The problem setting defines local subdomains, symmetric bilinear forms, and a coarse-space decomposition of the global function space. Local generalized eigenfunctions provide the basis for selecting coarse components while local contributions remain supported on overlapping subdomains.
- Geometric setting: The domain is partitioned by a quasiuniform quadrilateral or hexahedral triangulation with mesh parameter H, and overlapping patches are associated with its nodes.Each patch Ω_j is the union of cells surrounding a node x_j.
- Bilinear forms: The framework uses symmetric positive semidefinite local bilinear forms and a globally positive definite form on a separable Hilbert space V_0.On suitable local subspaces V_0(Ω_j), the local forms are required to be positive definite.
- Stable decomposition: The target decomposition represents every global function as one coarse component plus local components supported in the overlapping patches.The resulting additive Schwarz condition-number bound depends on the decomposition stability constant and the maximal overlap count.
- Local construction: Partition-of-unity functions define local patch interactions and ensure that localized products belong to the relevant global and local spaces.The maximal number of overlapping patches is denoted n_I, and the associated local bilinear forms are constructed using these functions.
- Local eigenproblems: For each patch, a generalized eigenvalue problem supplies eigenfunctions that are orthogonal in both local energy and mass-like inner products.Every local function can be expanded in these eigenfunctions, enabling projection onto the first L_j modes for coarse-space construction.
3. Coarse Spaces yielding Robust Stable Decompositions
The coarse space is built from selected local generalized eigenfunctions and combined with overlapping local components to obtain a stable decomposition. Under the stated assumptions, the resulting energy bound depends on overlap and threshold quantities rather than directly on mesh or coefficient contrast.
- Coarse-space construction: The coarse space is specified by patching selected local generalized eigenfunctions, with the threshold determining how many modes are retained from each patch.The construction uses the first L_j eigenfunctions associated with each local generalized eigenproblem.
- Assumptions: The abstract theory assumes symmetric positive semidefinite local forms, a globally positive definite form, positive definite local restricted forms, suitable partition-of-unity functions, and a threshold condition.These assumptions support the local and coarse energy estimates used in the stable decomposition.
- Local estimate: The local components satisfy an energy estimate whose constant depends only on the maximal number n_I of overlapping subdomains.The estimate is obtained using the overlap structure and Schwarz-type inequalities.
- Coarse estimate: The coarse component is bounded separately in the global energy norm, with its estimate controlled by the overlap count and the eigenvalue threshold.This estimate is combined with the local bound to control the complete decomposition.
- Combined result: Theorem 3.4 combines the local and coarse estimates to establish the stable decomposition under assumptions (A1)–(A5).The resulting constant is stated to depend only on n_I in the supplied theorem passage.
4. Applications
The abstract framework is applied to scalar elliptic, Darcy, Stokes, and Brinkman formulations, yielding stable decompositions and robust additive Schwarz preconditioners. Local eigenvalue structure controls coarse-space dimension and robustness against mesh size and coefficient contrast.
- Abstract framework: The framework gives condition-number bounds depending only on nI and τλ once assumptions (A1)–(A5) are verified.Choosing τλ independently of targeted problem and mesh parameters makes the bound independent of those parameters.
- Scalar elliptic case: For the scalar elliptic Galerkin problem, the relevant lower eigenvalue bound is uniform in κmax/κmin and H.This verifies (A5) with τλ independent of contrast and coarse-mesh size.
- Scalar elliptic case: The number of generalized eigenvalues below the threshold is bounded by the number of connected components in Ωs.The resulting coarse space therefore reflects the topology of high-conductivity regions.
- Darcy mixed formulation: The stream-function formulation of Darcy’s problem admits a stable decomposition and additive Schwarz preconditioner robust in κmax/κmin and H.A corresponding Darcy velocity-space coarse space is obtained by applying ∇× to coarse stream basis functions.
- Stokes formulation: For Stokes’ equations, the decomposition and additive Schwarz preconditioner are robust with respect to H.The analysis establishes a lower eigenvalue bound independent of H and transfers robustness to the equivalent velocity formulation.
- Brinkman formulation: For Brinkman’s problem, the decomposition and additive Schwarz preconditioner are robust with respect to H and κmax/κmin.As in the Darcy and Stokes cases, an equivalent robust preconditioner follows for the original formulation.
5. Reducing the Dimension of the Coarse Space
The section reduces the coarse-space dimension by replacing standard partition-of-unity functions with multiscale functions and retaining eigenfunctions associated with locally unrepresentable features. The construction addresses connected high-conductivity components while preserving the assumptions needed for robust decomposition.
- Multiscale partition of unity: The reduced construction replaces the standard partition of unity with multiscale functions that solve local conductivity-harmonic problems.Each multiscale function satisfies −∇·(κ∇eξj)=0 in T with boundary data ξj on ∂T.
- Eigenvalue reduction: The reduced coarse space avoids asymptotically small eigenvalues associated with high-conductivity components that do not touch coarse-cell boundaries.Additional zero-average conditions are needed for connected components of the enlarged subdomain that contain no relevant high-conductivity component.
- Robustness and scope: Under assumption (eA), the required assumptions (A4) and (A5) are satisfied, with constants independent of contrast and mesh size but potentially dependent on subdomain geometry.The paper notes that rigorous L∞ estimates used to verify (eA) remain an open problem beyond its scope.
- Multiscale spectral coarse space: The multiscale spectral coarse space is built by patching selected local eigenfunctions using the modified multiscale partition of unity.The local eigenproblems use the modified bilinear form associated with the multiscale construction.
- Spectral enrichment: For interior subdomains, the constant function is a zero-eigenvalue mode, while nonzero small-eigenvalue modes encode features that cannot be represented locally.Thus the method enriches the multiscale coarse space only with locally unresolved features.
6. Numerical Results
Numerical experiments evaluate standard, multiscale, spectral, and multiscale spectral coarse spaces for elliptic, mixed, and Brinkman problems under varying coefficient contrast. Spectral constructions yield contrast-robust preconditioners, while multiscale basis functions can substantially reduce coarse-space dimension.
- Implementation limitation: The finite-element interpolation used in computations is practical but is not directly covered by the abstract framework and requires future rigorous analysis.The alternative projection is local but may require inverting local stiffness matrices and can be unnecessarily expensive.
- Scalar elliptic and mixed problems: Standard coarse spaces exhibit condition numbers and iteration counts that grow with increasing contrast, whereas spectral coarse spaces remain contrast-independent.This behavior is reported for the scalar elliptic Galerkin formulation and the mixed formulation.
- Mixed and Brinkman problems: Spectral coarse spaces remain robust for Brinkman’s problem, although the underlying saddle point system is more difficult to solve.The mixed and Brinkman experiments use eigenfunctions below the threshold 1/τλ = 0.5.
- Coarse-space dimension: As contrast increases, spectral coarse-space dimension grows until reaching a maximum, with the reported configuration entering an asymptotic regime near contrast 1e4.More eigenvalues fall below the fixed threshold at higher contrast, lowering the estimated condition number before the dimension stabilizes.
- Dimension reduction: The multiscale spectral coarse space reduces the largest dimension from 838 to 60 in one example by using a multiscale partition of unity.The paper motivates this reduction for problems solved multiple times, where coarse-space dimension is especially important.
- Dimension reduction: In another experiment, the multiscale spectral coarse space has dimension at most 293 while the fine space has dimension 66049, with contrast-robust condition numbers.This demonstrates a small coarse space relative to the fine discretization in the reported configuration.
7. Conclusions
The paper develops stable decompositions for symmetric positive definite operators and applies them across several important finite element problems. Numerical experiments are reported as consistent with the theory and demonstrate the method’s usefulness.
- The framework constructs stable decompositions for symmetric positive definite operators under general assumptions.
- Applications cover scalar elliptic equations in Galerkin and mixed formulations, Stokes’ equations, and Brinkman’s equations.
- The scalar elliptic Galerkin formulation includes a strategy for reducing the coarse-space dimension.
- Numerical experiments agree with the theory and demonstrate the usefulness of the proposed method.