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Variational Theory and Domain Decomposition for Nonlocal Problems
Burak Aksoylu, Michael L. Parks
TL;DR
The paper addresses the lack of domain-decomposition methods for nonlocal operators and the need to characterize their conditioning. It develops variational formulations and analyzes stiffness and Schur-complement conditioning; both matrices are weakly mesh-size dependent, with distinct δ-dependent rates.
Problem
Method
The paper develops nonlocal variational formulations for single- and two-domain problems and uses spectral equivalence and numerical studies to analyze stiffness and Schur-complement conditioning.
Source: §1 · p. 2, §5 · p. 12, §5.1 · p. 13Results
κ(K) varies with δ^-2 and κ(SΓ) with δ^-1, while both are only weakly dependent on mesh size.
Source: §7 · p. 19Takeaways & Limitations
The analysis and experiments characterize conditioning for nonlocal stiffness and Schur-complement matrices, including their mesh-size and horizon dependence.
Source: §7 · pp. 18–19The role of a coarse problem in the nonlocal setting remains unexplored.
Source: §7 · p. 19
Abstract
from arXiv · showhide
In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincar\'{e} inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.
1. Introduction
Classical substructuring methods address local problems, while this paper develops and analyzes their nonlocal counterpart. It establishes a variational and conditioning framework and studies a nonlocal two-domain formulation.
1. Introduction
- Classical substructuring solves for interface unknowns to decouple subdomains, but the established methods discussed here address local problems.The interface variable may be primal, yielding subdomain Dirichlet problems, or dual, yielding Neumann problems.Source: §1 · p. 1
- Scalable methods require interface condition numbers that do not grow, or grow only weakly, with the number of subdomains; preconditioning uses a coarse problem to propagate error globally.Source: §1 · p. 1
- To the authors’ knowledge, this is the first domain-decomposition work for nonlocal models, aiming to characterize how nonlocality affects scalability.Source: §1 · p. 2
- The paper develops a nonlocal variational formulation, proves well-posedness and a nonlocal Poincaré inequality, and establishes stiffness and Schur-complement condition-number bounds.It also proves equivalence between single-domain and two-domain boundary value problems and reports supporting numerical studies.Source: §1 · p. 2
2. Interpretations of the Operator L
The paper situates the operator L in nonlocal diffusion and solid mechanics, describing the associated physical interpretations and assumptions on the kernel C.
2. Interpretations of the Operator L
- The operator L is used in applications including species-population evolution and image processing; the article examines two contexts and assumes an integrable C.Source: §2 · p. 2
2.1. Nonlocal Diffusion Processes
- For nonlocal diffusion with p = 2, the integral term describes material arriving at a point from the support of C, while the corresponding departure rate is also specified.Source: §2.1 · p. 3
- In the diffusion interpretation, C is nonnegative, radial, continuous, positive within a ball of radius δ, and zero outside that ball.Source: §2.1 · p. 3
2.2. Nonlocal Solid Mechanics
- Peridynamics is a nonlocal continuum-mechanics reformulation for discontinuous deformations, with u as displacement and C as the stiffness, or micromodulus, tensor.The stated assumptions include integrability and positive definiteness near x; the horizon δ bounds the support of C, and symmetry conditions are specified for elastic materials.Source: §2.2 · p. 3
3. A Nonlocal Variational Formulation
The section formulates nonlocal boundary value problems weakly on the domain and its nonlocal boundary. It uses the resulting bilinear form to establish well-posedness under stated conditions.
3. A Nonlocal Variational Formulation
- The variational formulation is constructed on Ω together with its nonlocal boundary, and the paper introduces a function space and weak form for the problem.Source: §3 · pp. 3–4
- The weak formulation assumes a finite iterated integral and an antisymmetric kernel-displacement product, which permit an identity by Fubini’s theorem.Source: §3 · p. 4
- For scalar functions, coercivity, continuity, and a bounded linear functional yield well-posedness of the nonlocal boundary value problem by the Lax–Milgram lemma.Source: §3 · p. 4
- The integration limits account for the support of C; the integration domain has four dimensions in 2D and six in 3D.Source: §3 · p. 4
4. Nonlocal Spectral Equivalence
The section develops a nonlocal Poincaré inequality and energy bounds to analyze stiffness-matrix conditioning, then tests the predicted mesh- and horizon-size behavior numerically.
4.3. An upper bound for a(u, u)
- A direct upper estimate for a(u, u) requires no discrete-space assumption, so it also holds in infinite-dimensional function spaces.The upper bound is reported to be sharp.Source: §4 · p. 5, §4.3 · p. 9
4.1. Nonlocal Poincar´e inequality
- A strip-covering nonlocal Poincaré inequality establishes coercivity without assuming every interior point interacts directly with the nonlocal boundary.The authors note that this avoids the stringent condition (4.16).Source: §4.1 · pp. 6–8
4.2. Demonstration of explicit δ-dependence in the nonlocal Poincar´e inequality
- A 1D Taylor-expansion demonstration with C4 functions and homogeneous nonlocal Dirichlet data gives the strongest lower bound when λP ncr=c(Ω)δ^3.The scaled nonlocal bilinear form asymptotically converges to the local form.Source: §4.2 · pp. 8–9
4.4. The Conditioning of the Stiffness Matrix K
- For sufficiently small δ, spectral equivalence gives an h-independent upper bound for the stiffness-matrix condition number, but the bound is not tight as δ→0.In that limit, the associated local stiffness-matrix condition number varies with h^-2.Source: §4.4 · p. 10
4.5. Numerical Verification of Condition Number by a Finite Element Formulation
- The numerical verification uses a Galerkin finite-element formulation on the unit cube with zero Dirichlet data on the nonlocal boundary, and estimates conditioning by the largest-to-smallest eigenvalue ratio.The tested shape functions are piecewise constant or piecewise linear.Source: §4.5 · p. 11
4.5.1. Results in One Dimension
- In 1D, the condition number varies as δ^-2 and is only weakly h-dependent; piecewise constant and linear shape functions show apparently similar conditioning.The figures report identical plots for the two shape functions.Source: Figure 4.1 · p. 10, Table 4.1 · p. 10, §4.5.1 · p. 11
4.5.2. Results in Two Dimensions
- In 2D, the condition number again varies as δ^-2 and depends only weakly on h, with piecewise constant shape functions.The maximum and minimum eigenvalues scale as δ^2 and δ^4, respectively.Source: Figure 4.2 · p. 11, §4.5.2 · p. 12
5. A Nonlocal Two-Domain Problem
The nonlocal two-domain formulation partitions the domain into overlapping subdomains sharing an interface and adjacent segments. Its weak form, including nonlocal transmission conditions, is equivalent to the original single-domain formulation.
5. A Nonlocal Two-Domain Problem
- The overlapping subdomains include each subdomain, the shared interface, and an adjacent segment, with the nonlocal boundary defined as a δ-thick region around Ω.The interface Γ lies between the overlapping open subdomains.Source: §5 · p. 12
5.1. Two-Domain Variational Form
- The construction uses a reduced bilinear form to represent the two-domain formulation.The test functions are supported only in the corresponding subdomains, avoiding coupling through the interface term.Source: §5.1 · p. 13
- The two-domain weak form uses subdomain equations and an interface transmission condition, with extension operators mapping interface data into each subdomain space.The test functions vanish on Γ, so each subdomain condition involves only that subdomain.Source: §5.1 · pp. 12–13
- The two-domain weak formulation is equivalent to the original single-domain formulation.The proof derives the transmission condition by partitioning the integrals and using extensions of interface data.Source: §5.1 · pp. 12–14
6. Towards Nonlocal Substructuring
The section develops a discrete two-domain formulation and a nonlocal Schur complement, then analyzes its conditioning theoretically and numerically. In 1D and 2D, the Schur-complement condition number depends weakly on mesh size and varies approximately as δ^-1.
6. Towards Nonlocal Substructuring
- The section constructs discrete subdomain equations and transmission conditions from the two-domain weak form, then introduces a nonlocal Schur complement for their interface problem.Numerical experiments are used to examine the complement’s conditioning.Source: §6 · p. 15
6.1. Linear Algebraic Representations
- The two-domain weak form is discretized using a finite element space on each subdomain and an interface space Λh, producing a block arrowhead global stiffness matrix.The first two block rows come from the subdomain equations, and the last from the transmission condition.Source: §6.1 · p. 15
6.2. Discrete Energy Minimizing Extension and the Schur Complement Conditioning
- The discrete energy-minimizing extension maps interface data into each subdomain, and its associated bilinear form corresponds to that subdomain’s Schur complement matrix.The extension minimizes subdomain energy among functions with the same interface values.Source: §6.2 · pp. 15–16
- Theorem 2 establishes a spectral equivalence for the discrete Schur complement, yielding a condition-number bound no greater than that of the corresponding stiffness matrix.The authors report that this estimate is not tight and that numerical experiments show smaller Schur-complement condition numbers.Source: §6.2 · p. 16
6.2.1. The Nonlocal Schur Complement Matrix
- The nonlocal Schur complement SΓ combines the two subdomain contributions and determines the interface solution.The section examines its conditioning as a function of h and δ.Source: §6 · p. 15, §6.2.1 · p. 16
6.3.1. Results in One Dimension
- In 1D, the condition number of SΓ varies nearly as δ^-1, compared with δ^-2 for the stiffness matrix K, and depends only weakly on h.The minimum and maximum eigenvalues depend linearly on h with a slope near unity; the reported plots show the same behavior for piecewise constant and piecewise linear shape functions.Source: §6.3.1 · p. 17, Figure 6.1 · p. 17, Table 6.1 · p. 17
6.3.2. Results in Two Dimensions
- In 2D, the condition number of SΓ varies as δ^-1 and depends only weakly on h.The minimum and maximum eigenvalues depend linearly on h with a slope of approximately two; the numerical study uses piecewise constant shape functions.Source: §6.3.2 · p. 18, Figure 6.2 · p. 18, Table 6.2 · p. 18
7. Conclusions and Future Work
The conclusions bring together the paper’s variational theory, domain-decomposition formulation, and conditioning results. They identify nonlocal coarse problems and other domain-decomposition methods as directions for future work.
Conclusions
- The paper constructs a nonlocal domain-decomposition framework with transmission conditions and proves equivalence between the one-domain and two-domain Dirichlet problems.It also defines an energy-minimizing extension to analyze the nonlocal Schur complement.Source: §7 · p. 18
- The paper proves well-posedness for nonlocal Dirichlet problems and a spectral equivalence yielding a mesh-size-independent upper bound for the stiffness-matrix condition number.The bound explicitly quantifies dependence on δ and dimension; the reported numerical experiments demonstrate sharpness of the component estimates.Source: §7 · p. 18
- Numerical results show that κ(K) and κ(SΓ) depend weakly on mesh size and vary with δ^-2 and δ^-1, respectively.The comparison with the local discrete Laplace equation notes h^-2 scaling for its stiffness-matrix condition number and h^-1 for its Schur-complement condition number.Source: Table 7.1 · p. 18, §7 · p. 19
Future Work
- The role of a coarse problem in the nonlocal setting remains unexplored, alongside convergence analysis and extensions to other domain-decomposition methods.The paper also identifies stable decompositions and local solvers as concepts that need to be reconstructed for nonlocal problems.Source: §7 · p. 19