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The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator
Marta D'Elia, Max Gunzburger
TL;DR
Fractional Laplacian problems on bounded domains are difficult when boundary traces or infinite-volume constraints are problematic. The paper uses a nonlocal formulation, proves convergence to fractional Laplacian solutions as interactions become infinite, and demonstrates numerical viability through finite element discretizations and examples.
Problem
Fractional differential equations on bounded domains can involve infinite-volume constraints, while boundary traces are not well defined for s ≤ 1/2.
Method
The paper treats the fractional Laplacian as a special case of a nonlocal diffusion operator, using nonlocal vector calculus, weak formulations, and continuous Galerkin finite element discretizations.
Results
Theoretical analysis and numerical examples show that nonlocal solutions converge to fractional Laplacian solutions as nonlocal interactions become infinite and provide viable approximations on bounded domains.
Takeaways & Limitations
Volume-constrained nonlocal diffusion problems offer a viable way to approximate fractional Laplacian problems on finite domains without treating infinite-volume constraints.
Takeaways & Limitations
The nonuniform-grid approach lacks a bound and an optimal coarsening algorithm, so its accuracy depends on empirical parameter tuning.
Abstract
from arXiv · showhide
We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem. We demonstrate that, when sufficient conditions on certain kernel functions hold, the solution of the nonlocal equation converges to the solution of the fractional Laplacian equation on bounded domains as the nonlocal interactions become infinite. We also introduce a continuous Galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates. Through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite-volume constraints.
1 Introduction and motivation
The paper develops bounded-domain methods for a nonlocal diffusion operator that includes the fractional Laplacian as a limiting special case. It uses nonlocal volume constraints, weak formulations, discretization analysis, and numerical comparisons to approximate fractional differential problems.
- Nonlocal diffusion operators model interactions at a distance and arise in applications including image analysis, machine learning, phase transitions, and mechanics.
- The paper develops discretization methods for fractional Laplacian equations on bounded domains by exploiting their relation to the nonlocal operator L.
- The fractional Laplacian is the limit of L as nonlocal interactions become infinite, provided sufficient conditions hold for the kernel functions.
- The paper excludes fractional Laplacian problems on the unbounded domain Ω = R^n and focuses strictly on bounded domains.
- Volume constraints generalize boundary conditions and enable bounded-domain treatment for s ≤ 1/2, where traces in the associated fractional energy space are not well defined.
- Nonlocal vector calculus provides a weak formulation used to prove convergence of nonlocal solutions to fractional differential solutions.
2 Elements of a nonlocal vector calculus
The paper introduces the nonlocal calculus underlying its diffusion operator, including nonlocal differential operators, interaction domains, kernels, and volume-constrained energy spaces. Under stated geometric and kernel assumptions, these spaces connect to fractional Sobolev spaces.
- The nonlocal divergence and gradient are adjoint operators, and L is defined as their composition with a symmetric positive definite tensor.
- The interaction domain ΩI contains points outside Ω that interact with points inside Ω; when ΩI = R^n\Ω, all exterior points may interact.
- The domain Ω, interaction domain ΩI, and their union are assumed bounded with piecewise smooth boundaries satisfying the interior cone condition.
- The kernel is symmetric, nonnegative, and supported within an interaction radius λ, making ΩI a layer of thickness λ surrounding Ω.
- The operator framework allows all s ∈ (0,1), and the nonlocal operator has broader applicability than fractional Laplacian problems alone.
- The volume-constrained energy space consists of energy-space functions that vanish on ΩI.
- For kernels satisfying the stated bounds, the nonlocal energy space is equivalent to the fractional Sobolev space H^s(Ω ∪ ΩI).
3 Relations between the nonlocal Laplacian and the fractional Laplacian
The fractional Laplacian is represented as a special case of the nonlocal diffusion operator, and nonlocal solutions converge to fractional-Laplacian solutions on bounded domains as the interaction radius grows.
- For kernels proportional to 1/|y − x|^(n+2s), the nonlocal operator becomes the fractional Laplacian.
- As λ → ∞, the solution of the volume-constrained nonlocal problem converges to the solution of the fractional-Laplacian problem.
- The error bound scales as C1^s(λ − I)^(2s) times the L2 norm of the fractional-Laplacian solution.
- Lower s produces slower convergence, while the L2 error estimate is reported to be sharp in numerical illustrations.
4 Finite-dimensional approximations
The paper develops finite-dimensional and finite element approximations for the nonlocal problem, analyzes their convergence, and addresses computational costs from large interaction domains using nonuniform grids.
- 4 Finite-dimensional approximations: Finite-dimensional Galerkin solutions converge to the nonlocal solution as the approximation-space dimension N tends to infinity.
- 4 Finite-dimensional approximations: Continuous finite element spaces use piecewise polynomials on shape-regular, quasi-uniform partitions, with N increasing as the mesh size h decreases.
- 4 Finite-dimensional approximations: The total approximation error combines the interaction-radius error with the finite element discretization error.
- 4 Finite-dimensional approximations: For very large λ, uniform-grid computation becomes unaffordable, motivating coarser grids farther from Ω where integral contributions are less significant.
- 4.2 Nonuniform grid in ΩI: No bound or optimal coarsening algorithm is available; empirically tuned coarsening can preserve accuracy, while quadrature and grid coarsening may introduce integration error.
5 Numerical tests
The numerical tests evaluate finite element approximations of nonlocal solutions and their convergence toward fractional-Laplacian solutions as grid resolution and interaction radius increase. They also examine nonuniform grids and show that convergence rates depend on both discretization error and the fractional order.
- Finite element discretization: The one-dimensional tests use continuous piecewise-linear finite elements on partitions of Ω∪ΩI, with standard hat functions and numerical quadrature for the nonlocal integrals.The nonlocal stiffness matrix is dense, making computations more costly than in the local case.
- Finite element discretization: Uniform-grid computations become expensive for large interaction radii because the nonlocal stiffness matrix is dense and interactions span much of Ω∪ΩI.Coarser grids in ΩI are therefore considered to reduce computational cost.
- Test problems: The analytic test solutions include cases with s below and above 1/2, and the corresponding solutions are used to assess finite element and fractional-Laplacian convergence.Analytic solutions are available for the ball problem through a Green's-function representation.
- Nonuniform grids: For p<0.5, the nonuniform-grid coarsening error is negligible in the reported N=27 tests.The parameter p controls coarseness, with larger p producing a coarser grid.
- Convergence to the fractional Laplacian: For s>0.5, increasing both λ and N brings the numerical nonlocal solutions closer to the fractional-Laplacian solution u.This behavior is reported for test cases Ia and IIa, including peak-focused comparisons.
- Convergence to the fractional Laplacian: For s<0.5, increasing λ reduces solution amplitude while increasing N raises it, and simultaneous increases produce convergence to u from above.A very fine grid would be required to observe the conjectured fixed-grid λ-convergence rate directly.
- Convergence to the fractional Laplacian: The observed convergence rate is 0.5 when λ and N increase together, indicating that finite element approximation error dominates convergence toward u.For fixed N, convergence with respect to λ follows the predicted -2s rate in both energy and L2 norms against a fine-grid surrogate.
6 Concluding remarks
The paper concludes that nonlocal diffusion solutions converge to fractional Laplacian solutions as interactions become infinite, and that numerical examples support using nonlocal problems as approximations on bounded domains. It also identifies extensions to time-dependent problems, higher-dimensional simulations, improved finite-element convergence, and whole-space fractional Laplacian problems.
- Nonlocal diffusion solutions converge to fractional Laplacian solutions as nonlocal interactions become infinite.
- Finite-element convergence could be improved by incorporating the fractional Laplacian solution’s endpoint singularity through supplementary singular basis functions.The analytic solution has a singular derivative at the domain endpoints, which limits the observed grid-convergence rate.
- A time-dependent nonlocal diffusion equation is proposed for future study, including applications to jump processes and possible fractional time derivatives.The time-dependent equation is described as governing the evolution of a jump process’s probability density on a bounded domain.
- Whole-space fractional Laplacian problems remain outside the study’s scope and would require examining finite-domain approximations as the computational domain grows.
- For s ≤ 1/2, discontinuous Galerkin discretizations are a proposed extension because both problem types can admit jump discontinuities.
- Numerical results illustrate the theory and support volume-constrained nonlocal diffusion as an approximation strategy for bounded-domain fractional Laplacian problems.The numerical results are described as preliminary, with higher-dimensional extensions expected to be more computationally challenging.