Numerical methods for nonlocal and fractional models

Marta D'Elia; Qiang Du; Christian Glusa; Max Gunzburger; Xiaochuan Tian; Zhi Zhou

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Numerical methods for nonlocal and fractional models

Marta D'Elia, Qiang Du, Christian Glusa, Max Gunzburger, Xiaochuan Tian, Zhi Zhou

arXiv:2002.01401v2Feb 4, 2020math.NAmath.APmath.OC

TL;DR

The paper develops numerical approaches for nonlocal diffusion models that represent physics distinct from local PDE formulations. It analyzes finite element, adaptive, spectral, and fractional-model approximations, including convergence, error estimation, and scope limitations.

Problem
Nonlocal models are needed when the modeled physics differs from a local PDE formulation, including interactions represented over finite distances.
Method
The paper studies finite element, residual-based adaptive refinement, and spectral methods, while relating many fractional PDEs to limiting nonlocal models.
Results
Quadratic L2 convergence is established for the spectral-Galerkin approximation uN,0 to the local solution u0 under f ∈ L2(−π, π), alongside reliable and efficient residual error estimators for scalar nonlocal problems.
Takeaways & Limitations
Many widely studied fractional PDEs can be treated as specialized nonlocal models or limiting cases, extending the applicability of the nonlocal framework.
Takeaways & Limitations
Optimal convergence without cut-off functions remains open for the radial basis function method, while the presented convection–diffusion analysis is effective for diffusion-dominated problems.

Abstract

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Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to adequately model observed phenomena or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference, and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modeling and algorithmic extensions which serve to show the wide applicability of nonlocal modeling.

PART ONE

The article introduces nonlocal diffusion models as integral formulations in which points interact across a finite horizon, and develops the nonlocal calculus used to analyze and discretize them.

PART ONE: Nonlocal models describe interactions at a distance through integral equations rather than PDE formulations.The article presents them as models of different physics, not merely integral reformulations of local PDEs.
PART ONE: The classical Poisson model uses boundary conditions, whereas the nonlocal analogue imposes volume constraints on an interaction domain.The interaction domain contains exterior points within distance δ of the physical domain.
PART ONE: The horizon δ controls which points interact, with regimes ranging from δ much smaller than diam Ω to δ = ∞.The interaction domain changes with the horizon and may extend over the entire exterior when δ is infinite.
PART ONE: The generic nonlocal model uses −Lδu = f in Ω and Vu = g in ΩIδ, with Dirichlet, Neumann, Robin, or mixed volume constraints available.The kernel vanishes beyond distance δ, and Lδ is the nonlocal analogue of ∇·(D∇u).
PART ONE: A nonlocal vector calculus supplies integral operators analogous to divergence, gradient, and curl for modeling and analysis.Its operator definitions depend on the horizon and antisymmetric kernel αδ, while compositions can generalize constant-coefficient operators to variable coefficients.

2. Weak formulations of nonlocal models

This section formulates nonlocal diffusion variationally, establishes well-posedness through energy spaces and coercive bilinear forms, and distinguishes operator kernels from constitutive functions.

2. Weak formulations of nonlocal models: The strong nonlocal problem is recast in a weak formulation using constrained energy spaces and a bilinear form.The formulation prescribes u = g on the interaction domain and tests against functions vanishing there.
2.1.1. Function spaces and norms: The energy space V(Ω∪ΩIδ) is a subspace of L2(Ω∪ΩIδ) and, for suitable kernels, is a Hilbert space.Constrained spaces and trace-like spaces are induced by the nonlocal energy norm.
2.1.1. Function spaces and norms: Under suitable kernel conditions, a nonlocal Poincaré inequality makes the constrained seminorm an equivalent norm with constants independent of δ as δ → 0.This supports the functional-analytic treatment of the constrained problem.
2.1.2. Weak formulations and well-posedness: The bilinear form is symmetric, continuous, and coercive, so the Lax–Milgram theorem gives well-posedness and an a priori energy estimate.For suitable kernels, the same framework also supports local and fractional limits.
2.1.2. Weak formulations and well-posedness: Nonlocal Neumann formulations require quotient spaces for uniqueness and compatibility conditions on the data for existence.Alternative Neumann operators and fractional variants are also discussed.
2.1.3. Relation to an energy minimization principle: The weak problem is the Euler–Lagrange equation of an energy minimization principle under suitable kernel and data conditions.This connects variational formulations with minimization of a nonlocal energy functional.
2.2. Kernel choices and the corresponding energy spaces: The kernel representation separates operator-defining and constitutive roles: φδ relates to αδ, while θδ relates to Θδ.The model assumes nonnegative symmetric scalar functions and an antisymmetric directional kernel structure.
2.2.1. A list of kernel functions in common use: The listed kernel functions determine corresponding energy spaces, while scalar constitutive tensors and radial functions provide common special cases.The article focuses on selected kernel functions rather than every square-integrable possibility.

3. Fractional diffusion models

The article treats integral and spectral fractional Laplacians as nonlocal diffusion models, analyzes their variational structure and limits, and discusses truncation and analytic solution behavior.

3. Fractional diffusion models: Fractional diffusion uses the fractional Laplace operator −(−∆)^s with s ∈ (0, 1), corresponding to a specific nonlocal kernel choice.The fractional Laplacian is also associated with superdiffusion and heavy-tailed Lévy flights.
3. Fractional diffusion models: The integral fractional Poisson problem imposes (−∆)^s u = f in Ω and u = g on the exterior interaction domain.It is the restriction of the full-space operator subject to a volume constraint.
3. Fractional diffusion models: As s → 1−, the integral fractional diffusion solution strongly converges to the local diffusion solution in H1−ϵ(Ω).The integral representation is equivalent to a Fourier representation, and the operator approaches −∆ as s → 1−.
3. Fractional diffusion models: The regional fractional Laplacian changes both the bounded-domain operator and its integration domain compared with the integral fractional Laplacian.The integral fractional operator integrates over Rd, whereas the regional operator integrates over Ω.
3. Fractional diffusion models: The variational treatment uses fractional Sobolev spaces, Green’s identities, and a coercive continuous bilinear form to establish weak solvability.Homogeneous and inhomogeneous volume constraints can be incorporated directly or enforced with Lagrange multipliers.
3. Fractional diffusion models: For nonzero exterior data, computation can avoid integrating over the unbounded exterior except for a data integral that may be approximated by quadrature.Homogeneous exterior data restrict the right-hand side to integrals over Ω.
3. Fractional diffusion models: Finite-horizon truncations with δ > diam(Ω) provide approximations to the infinite-horizon problem and can also model practical large but finite interactions.The truncated problem uses the same source and volume data on its finite interaction domain.
3. Fractional diffusion models: The spectral fractional Laplacian is defined from eigenvalues and eigenfunctions of the PDE Laplacian with homogeneous Dirichlet conditions.It recovers the identity as s → 0 and the integer-order Laplacian as s → 1.

4. Introductory remarks

The introductory remarks identify coupled horizon and discretization limits as a central numerical issue and motivate asymptotically compatible schemes.

4. Introductory remarks: Nonlocal discretizations involve both a horizon δ and a discretization parameter h or spectral dimension N.Continuous and discrete models therefore have multiple limiting behaviors to analyze.
4. Introductory remarks: The order in which δ and discretization limits are taken can change the resulting limit.The article therefore seeks schemes whose limiting behavior is not sensitive to the path through parameter space.
4. Introductory remarks: Consistency with local PDE models may require mesh or quadrature spacing to decrease faster than the horizon parameter.Proportional coupling can produce inconsistent limiting solutions.
4. Introductory remarks: Asymptotically compatible schemes are designed to avoid inconsistent limits while allowing efficient choices of model and discretization parameters.The framework compares continuous and discrete local and nonlocal solutions as parameters vary.
4. Introductory remarks: When δ is proportional to h, piecewise constant conforming finite element and simple Riemann-sum approximations can converge to an incorrect limit.These failures motivate alternative AC discretizations for nonlocal models.

5. Finite element methods for nonlocal models

Finite element methods for nonlocal models adapt standard Galerkin ideas to interaction domains and nonlocal energy spaces, while asymptotic compatibility governs correct local and fractional limits. Adaptive estimators and meshes address computational cost and solution discontinuities.

Finite element formulation: Finite element discretization subdivides the interaction domain, constructs a discrete energy space, approximates volume data, and solves a coercive Galerkin problem.Singular kernels require specialized quadrature for stiffness-matrix evaluation.
Asymptotic compatibility: Continuous piecewise linear conforming Galerkin spaces are asymptotically compatible and recover the correct local limit as both δ and h decrease.The result holds in any space dimension, even when δ decreases faster than h.
Asymptotic compatibility: Piecewise constant schemes require h = o(δ) for the correct local limit, whereas constant bandwidth can produce an incorrect limit.For general nonlocal systems, bandwidth must grow mildly as the mesh is refined.
Adaptive methods: Adaptive finite element methods use residual estimators and convergent refinement algorithms, but extensions to local limits, time dependence, and nonlinear models remain open.Reduced sparsity makes nonlocal computations more expensive than local PDE computations, motivating adaptive methods.
Adaptive methods: Elongated elements around discontinuities and abrupt transitions to regular elements reduce degrees of freedom while supporting h2 convergence.The mesh uses thickness O(h4) across discontinuities and length O(h) along them; the resulting degrees of freedom can be comparable to a regular local-PDE grid.
Fractional limits: Nonlocal models bridge local and fractional models, with δ →0 yielding the local limit and δ →∞ yielding the fractional limit under suitable conforming discretizations.Conforming Galerkin approximations can converge to the fractional solution as δ →∞ without requiring h to depend on δ.

6. Finite element methods for the integral fractional Laplacian

Finite element methods for the integral fractional Laplacian must accommodate limited solution regularity, boundary singularities, dense matrices, and singular element-pair integrals. Graded or adaptive meshes and specialized quadrature provide convergence while addressing these difficulties.

Finite element spaces: Piecewise linear finite element spaces depend on s: all nodal basis functions are used for s < 1/2, while boundary nodes are excluded for s ≥1/2.The discrete space is defined within the fractional energy space and solves the corresponding Galerkin problem.
Convergence and regularity: Boundary regularity limits convergence on quasi-uniform meshes, so boundary refinement can restore optimal rates in one dimension but not generally in higher dimensions.The stated higher-dimensional expectation is no more than O(N−1/2+ε) in d = 2 dimensions.
Quadrature rules: The integral fractional Laplacian produces a dense algebraic system whose stiffness entries involve singular integrals over element pairs and external edges.Non-disjoint element pairs require singular-integration techniques adapted from boundary element methods.
Quadrature rules: Non-uniform Gauss-type and transformed hypercube quadrature rules address singular interactions between element configurations.The quadrature analysis classifies identical, edge-sharing, vertex-sharing, and separated element pairs.
Quadrature rules: Using O(log N2d_h) quadrature points per element pair makes quadrature consistency error dominated by discretization error.This result applies to the quadrature approximation of the fractional bilinear form.

7. Finite element methods for the spectral fractional Laplacian

Finite element methods for the spectral fractional Laplacian use an extension problem on a semi-infinite cylinder and recover the solution by tracing the extended variable. Truncation and hybrid spectral approaches provide practical discretizations, with complexity remaining a central concern.

Extension formulation: The spectral fractional Poisson solution is recovered as the trace of an extension-problem solution posed on the semi-infinite cylinder Ω × [0, ∞).The extension formulation uses a weighted solution space and a corresponding weak form.
Truncation approach: One approach truncates the unbounded direction at ztrun and discretizes the resulting cylinder with finite elements; truncation introduces an exponentially small error.The truncated problem imposes a homogeneous Dirichlet condition at z = ztrun.
Hybrid spectral approach: A hybrid approach uses finite elements in the spatial direction and a spectral method in the extended direction, avoiding direct truncation of the semi-infinite domain.The spectral construction uses approximated eigenvalues and decimation, requiring only |log h|p eigenvalue approximations for some p > 0.
Truncation approach: A graded tensor-product mesh with ztrun ∼ |log N| yields a finite element approximation of the spectral fractional solution.The construction combines a quasi-uniform spatial mesh with a graded mesh in the extended direction.
Complexity: The truncation method is optimal in regularity but sub-optimal in complexity, motivating sparse-grid and hp-finite-element alternatives.The hp-finite-element approach can achieve exponential convergence, including for low domain regularity and incompatible forcing.
Linear-system solution: When the extension discretization has tensor structure, solving reduces to a sequence of integer-order reaction–diffusion problems handled by classical iterative solvers.Conjugate gradient and multigrid methods are identified as suitable solvers.

8. Spectral-Galerkin methods for nonlocal diffusion

The section develops Fourier spectral-Galerkin methods for periodic nonlocal diffusion, including eigenvalue computation and asymptotic compatibility. Extensions address spherical domains and nonlinear phase-field models, while highlighting Gibbs phenomena and boundary-condition limitations.

Periodic nonlocal diffusion: The method uses Fourier eigenfunctions e±inx, whose nonlocal eigenvalues approach the local values n2 as δ →0.
Periodic nonlocal diffusion: The Fourier spectral-Galerkin scheme is asymptotically compatible and admits uniform error estimates for sufficiently small δ and sufficiently large N.The estimates do not require a restriction on the relative sizes of δ and N.
Periodic nonlocal diffusion: The local Fourier spectral solution converges in L2 at least quadratically with respect to 1/N when f ∈ L2(−π, π).
Implementation: For singular kernels and highly oscillatory integrals, hybrid algorithms combine series truncation for small nδ with high-order Runge-Kutta integration for large nδ.
Extensions: On the sphere, spherical harmonics are eigenfunctions of the nonlocal Laplace–Beltrami operator, with eigenvalues computed using modified Clenshaw–Curtis quadrature and asymptotic formulas.The reported per-eigenvalue complexity decreases from O(l2) to O(l log l) for sufficiently large l.
Limitations: Spectral methods can capture discontinuities but may produce spurious Gibbs phenomena; current uniform convergence analyses focus on periodic boundary conditions.Extensions to Dirichlet or Neumann volume constraints remain open because comparable closed eigenvalue expressions are unavailable.

9. Spectral-Galerkin methods for fractional diffusion

The section surveys spectral methods for fractional diffusion on unbounded and bounded domains. Hermite, mapped Gegenbauer, and Jacobi-based methods address algebraic decay or boundary singularity, with convergence determined by solution regularity and domain geometry.

Unbounded domains: Fractional solutions on R^d decay algebraically, making naive domain truncation inaccurate and motivating global orthogonal-function approximations.Artificial boundary conditions for the fractional case remain largely open.
Unbounded domains: The Hermite-Galerkin method has a diagonal mass matrix and efficiently computable stiffness matrix, but its error estimate predicts algebraic rather than exponential convergence.The rate reflects algebraic decay of the solution or source term at infinity.
Unbounded domains: Mapped Gegenbauer bases are better suited than classical Hermite or Laguerre bases to functions with algebraic decay rates.
Unbounded domains: The mapped Gegenbauer approach extends efficiently to higher dimensions through tensor products, with complexity O(N(log N)^d).Sinc quadrature evaluates the transformed integration with negligible cost relative to an FFT.
Bounded domains: In bounded domains, low solution regularity limits classical spectral approximation, whereas weighted Jacobi formulations yield exponential convergence when the source term is smooth.The derived estimate is stated in a weighted L2-norm.
Limitations: Higher-dimensional bounded-domain fractional spectral analysis remains difficult because irregular-domain methods and regularity theory are not sufficiently developed.

10. Finite difference methods for the strong form of nonlocal diffusion

The section presents asymptotically compatible finite difference and reproducing-kernel collocation methods for the strong form of nonlocal diffusion. Uniform consistency and stability depend on carefully designed interpolation, weights, and basis properties.

Finite differences: Quadratic exactness makes the Cartesian finite difference scheme an O(h2) approximation independent of δ.This yields uniform truncation error for small δ.
Finite differences: Once stability is established, the finite difference solution approximates the local solution at rate O(h2 + δ2).
Finite differences: The finite difference scheme is second-order accurate and satisfies a discrete maximum principle because its coefficients are nonnegative.
Finite differences: Higher-order finite differences lose the discrete maximum principle when interpolation coefficients become negative, leaving stability analysis open except for specialized kernels.
RK collocation: Reproducing-kernel collocation achieves asymptotic compatibility through exact multilinear reproduction, quadratic shifting, and synchronized convergence.
RK collocation: The linear RK basis has a nonnegative Fourier transform, enabling comparison with a stable Galerkin approximation, but all RK collocation schemes fail the discrete maximum principle.

11. Numerical methods for the strong form of fractional diffusion

The section presents strong-form numerical methods for fractional diffusion, emphasizing quadrature-based finite differences, Monte Carlo simulation, and radial basis functions. It discusses singular-integral treatment, probabilistic representations, stability, efficiency, and convergence limitations.

11. Numerical methods for the strong form of fractional diffusion: Three strong-form approaches are considered: quadrature rule-based finite differences, Monte Carlo methods, and radial basis function methods.
11.1. Quadrature rule-based finite difference methods: Quadrature-based finite differences split the singular fractional-Laplacian integral so its singular and regular parts can be approximated separately.Window-function and alternative splitting strategies enable trapezoidal or analytic treatment of the resulting terms.
11.1. Quadrature rule-based finite difference methods: Maximum-principle arguments establish stability for the finite-difference scheme and yield error estimates, while higher-dimensional extensions improve convergence rates.
11.1. Quadrature rule-based finite difference methods: Uniform grids can produce translation-invariant operators suitable for FFT solution, whereas graded meshes create dense, unstructured matrices.Hierarchical matrices are proposed to reduce storage and computational complexity for unstructured meshes.
11.1. Quadrature rule-based finite difference methods: The one-dimensional fractional problem is discretized on a uniform Cartesian grid using integral splitting, Taylor expansion, central differences, interpolation, and quadrature.
11.2. Monte Carlo method by Feynman–Kac formula: The fractional Feynman–Kac representation replaces Brownian motion with an isotropic α-stable Lévy process that exits domains by jumps.Monte Carlo estimates use independently sampled paths and the law of large numbers.
11.2. Monte Carlo method by Feynman–Kac formula: Direct Feynman–Kac Monte Carlo is inefficient because evaluating a solution at one point requires simulating a very large number of paths.
11.3. Radial basis function methods: Radial basis function analysis imposes 2β < k and 2β−2s > n, but the optimal convergence rate without cut-off functions remains open.Experiments found no measurable difference between estimations with and without the smooth cut-off function.

12. Conditioning and fast solvers

The section relates nonlocal conditioning and solver design to both mesh size and interaction horizon. It reviews FFT, multigrid, fast multipole, hierarchical-matrix, and preconditioning strategies, while identifying dense matrices and non-smooth kernel truncation as computational challenges.

Nonlocal algebraic systems require conditioning analysis because efficient simulation depends on solving the discretization-generated systems effectively.
Nonlocal stiffness-matrix condition numbers generally depend on both the horizon δ and mesh size h, unlike the typical local diffusion scaling O(h−2).
Toeplitz and multigrid solvers are among the approaches studied for effective solution of nonlocal linear systems.
12.1. Fast algorithm for kernels with non-smooth truncation: Fast multipole methods decompose singular, finitely truncated kernels and compress interactions away from the support boundary, where the kernel is smooth.Near the geometric boundary, small boxes resolve the interaction kernel; farther away, larger boxes are used.
12.1. Fast algorithm for kernels with non-smooth truncation: The interaction region is represented geometrically and decomposed into hierarchical boxes, with a close-up showing finer resolution.
12.2. Conditioning and solvers for finite element discretizations of the integral fractional Laplacian model: Finite element stiffness matrices for the integral fractional Laplacian have analyzed spectra and condition numbers on shape-regular triangulations.
12.2. Conditioning and solvers for finite element discretizations of the integral fractional Laplacian model: Multigrid can restore a uniform iteration bound when the iteration count depends on problem size for larger fractional orders or steady-state problems.
12.2. Conditioning and solvers for finite element discretizations of the integral fractional Laplacian model: Dense nonlocal system matrices motivate fast transforms and matrix compression, both yielding quasi-optimal O(N_h log N_h) solve complexity under their respective conditions.Fast transforms are limited to uniform meshes, while matrix compression is more difficult to implement.

13. Weakly coercive, indefinite and non-self-adjoint problems

This section extends nonlocal discretization theory beyond symmetric coercive problems to indefinite and non-self-adjoint formulations. It uses boundedness and inf-sup conditions for well-posedness, develops asymptotic-compatibility requirements, and discusses mixed and operator-splitting formulations.

The section addresses indefinite and non-self-adjoint problems that cannot be treated by the Lax–Milgram theorem.
The generalized formulations use bilinear forms without symmetry or coercivity, allowing treatment of settings excluded by the symmetric coercive form.
Boundedness, an inf-sup condition, and a nondegeneracy condition guarantee well-posedness when the right-hand-side functional is bounded.
Conforming discrete trial and test spaces must satisfy approximation, discrete inf-sup, and related requirements to support well-posed discrete problems.
Independence of stability constants from δ is imposed for asymptotic-compatibility analysis, although it is unnecessary when solving at a fixed δ.
The framework includes nonlocal mixed, operator-splitting, convection–diffusion, peridynamic, and nonlocal Stokes settings.
Asymptotic compatibility additionally requires discrete test spaces to become asymptotically dense in the local PDE energy space as δ and h approach zero.
Operator splitting mimics the local balance-law and constitutive-law decomposition, while mixed discretizations require approximating two-variable fields in both x and y.

14. Nonlocal convection–diffusion problems

The paper develops fully nonlocal convection–diffusion models in which both diffusion and convection are represented nonlocally, producing generally non-symmetric kernels. It gives their operator structure, weak formulation, well-posedness framework, and scope conditions for the analysis.

14. Nonlocal convection–diffusion problems: Fully nonlocal models replace both the diffusion and convection terms, yielding non-symmetric kernels that can represent non-symmetric diffusion.The diffusion and convection contributions may be treated as separate phenomena in a general operator framework.
14. Nonlocal convection–diffusion problems: The general operator allows different horizons and kernel functions for diffusion and convection through separate nonlocal divergence operators.The nonlocal diffusion tensor and convection velocity need not produce radial or translationally invariant kernels.
14. Nonlocal convection–diffusion problems: Specialized formulations include conservative operators with indicator functions depending on interaction distance and the velocity field.These operators are generically non-symmetric and can be viewed as special cases of the general nonlocal convection–diffusion operator.
14.1.1. Steady-state nonlocal convection–diffusion problems: Weak formulations define constrained energy spaces, bilinear forms, and linear functionals for analyzing steady-state nonlocal convection–diffusion problems.The framework assumes a nonlocal Poincaré inequality and places the forcing in the dual space of the energy space.
14.1.1. Steady-state nonlocal convection–diffusion problems: The stated analysis is effective for diffusion-dominated problems, while convection-dominated cases require a more general formulation.The solution estimate involves the coercivity constant of the associated bilinear form.

15. Time-dependent nonlocal problems

The article extends nonlocal diffusion, convection–diffusion, fractional, and optimal-control models to time-dependent settings, with weak formulations and discretizations. It records decay, dispersion, convergence, regularity, and approximation results, while emphasizing differences from local PDEs.

15. Time-dependent nonlocal problems: Time-dependent nonlocal problems can use steady-state spatial discretizations together with standard temporal discretizations.The article notes that spatial discretization can use finite element, finite difference, or spectral methods developed for steady-state problems.
15.1. Time-dependent nonlocal diffusion: Nonlocal diffusion solutions satisfy diffusive decay properties but may remain no smoother than the data for radial, integrable kernels.This contrasts with the smoothing commonly associated with parabolic PDEs.
15.1. Time-dependent nonlocal diffusion: The fractional heat equation extends the time-dependent framework using the fractional Laplacian and prescribed exterior values.The model includes an initial condition on the domain and exterior data outside it.
15.2. Time-dependent convection–diffusion problems: Weak formulations of time-dependent nonlocal convection–diffusion problems are well-posed under coercivity, continuity, and the weaker condition ∥Dc,δµ∥∞ < ∞.The latter condition concerns the nonlocal convection velocity µ(x, y).
15.3. Nonlocal wave models: Nonlocal wave velocity depends nonlinearly on wavenumber, and converges quadratically in δ to the local wave velocity as δ → 0.The same type of dispersion result is also reported in two dimensions.
16. Nonlocal optimal control: Finite element analyses establish convergence and error estimates for time-dependent optimal-control approximations, with numerical illustrations in one and two dimensions.Related results include semidiscrete and fully discrete schemes with state and control error estimates.

17. Variational inequalities and obstacle problems

Nonlocal obstacle problems are formulated as variational inequalities and analyzed through well-posedness, regularity, mixed finite element discretization, and numerical examples. Compared with local obstacle problems, they permit less smooth obstacles and can produce substantially different multipliers and solution behavior.

17. Variational formulation: The nonlocal obstacle problem imposes an operator inequality, obstacle constraint, complementarity condition, and homogeneous volume constraint.Its mixed formulation seeks a state u and multiplier λ in a closed convex dual cone.
17. Analysis: Nonlocal obstacle problems require less smooth obstacles for well-posedness than corresponding local PDE obstacle problems.The article also reports that local and nonlocal solution behavior can differ substantially.
17. Analysis: For fractional Laplacian obstacle problems, improved regularity holds for every s ∈ (0, 1), with β = min{s, 1/2 − ε}.The stated assumptions include f ∈ L2(Ω) and (−Lδψ − f)+|Ω ∈ L2(Ω).
17. Finite element methods: Piecewise linear continuous primal and discontinuous multiplier finite element spaces yield a well-posed discrete problem through inf-sup stability.Locally bi-orthogonal basis functions are used for the discontinuous multiplier space.
17. Numerical example: Nonlocal and local obstacle models produce different multipliers: λ belongs to L2(Ω) nonlocally, whereas the local multiplier consists of Dirac delta functions.These differences are illustrated alongside differences in the primal solutions.
17. Fractional kernels: Results proved for the fractional Laplacian also cover truncated fractional kernels, whose obstacle solutions converge to those of the un-truncated operator.This extends the theoretical analysis beyond the un-truncated fractional kernel.

18. Reduced-order modelling

Reduced-order modelling constructs low-dimensional approximations for parametrized nonlocal problems, motivated by the size and reduced sparsity of nonlocal discrete systems. The examples show that increasing the POD basis dimension can substantially improve approximation quality.

18. Reduced-order modelling: ROM constructs a very low-dimensional discretization from high-dimensional discretizations for parametrized problems.Its construction typically incurs an offline cost from selected high-dimensional runs, after which the ROM supports online simulations.
18. Reduced-order modelling: Nonlocal discretizations can be large and much less sparse than PDE discretizations, and fractional-Laplacian systems may be full.This combination creates a high-dimensionality and lack-of-sparsity burden for computation.
18. Reduced-order modelling: The ROM setting uses parameter vectors for both the constitutive function and kernel, then variationally discretizes the resulting nonlocal diffusion problem.The high-dimensional discretization has dimension Nh, while the ROM seeks dimension Nrom much smaller than Nh.
18. Reduced-order modelling: Greedy reduced bases and proper orthogonal decomposition are established approaches for reduced-order models of nonlocal diffusion.The cited examples include parameter dependence through the constitutive function and, in later work, through the horizon and fractional exponent in the kernel.
18. Reduced-order modelling: Six POD basis functions closely approximate the high-dimensional discontinuous Galerkin solution, whereas one POD basis function performs poorly.The comparison uses a singular, non-integrable kernel, a discontinuous manufactured solution, and five time instants.
18. Reduced-order modelling: ROM solves depend on Nrom, but naive ROM assembly still contains steps whose cost depends on the high-dimensional dimension Nh.Strategies developed for PDE ROMs can also address this assembly-cost obstacle in the nonlocal setting.

20. Peridynamics models for solid mechanics

Peridynamics provides a nonlocal continuum model for vector-valued displacement in solid mechanics, formulated through energy minimization and capable of representing defects such as fractures. Its conforming Galerkin discretizations can recover the classical local elasticity limit under suitable refinement conditions.

20. Peridynamics models for solid mechanics: Peridynamics is a nonlocal continuum model for vector-valued displacement that provides an analogue of the classical Navier equations of linear elasticity.The model discussed is the linear state-based formulation introduced and analyzed by Mengesha and Du.
20. Peridynamics models for solid mechanics: The peridynamics problem is defined by an energy minimization principle whose Euler–Lagrange equation gives a weak formulation.The formulation uses nonlocal operators, an energy space, and a symmetric bilinear form involving material constants related to bulk and shear moduli.
20. Peridynamics models for solid mechanics: Coercivity and continuity of the peridynamic bilinear form establish well-posedness through the Lax–Milgram theorem.The result is stated for the energy space Vp,δ and extends to discussions involving other nonlocal constraints.
20. Peridynamics models for solid mechanics: Peridynamic solutions can admit jump discontinuities, allowing displacement discontinuities to represent fractures; time-dependent models may add bond-breaking rules for defect nucleation.A bond is described as becoming un-bonded when initially neighboring points later separate by more than the horizon distance δ.
20. Peridynamics models for solid mechanics: Finite element discretization seeks uδ,h in a finite element subspace satisfying the peridynamic bilinear-form equation for every test function in that subspace.The mesh size h controls approximation of the energy-space solution for fixed δ.
20. Peridynamics models for solid mechanics: Conforming Galerkin approximations containing continuous piecewise linear functions are automatically asymptotically compatible and recover the correct local limit as δ and h decrease.For discontinuous piecewise constants, recovery can require h = o(δ); fixed bandwidth during refinement may instead produce an incorrect local limit.