What Is the Fractional Laplacian?

1.1. Overview

• On bounded domains, equivalent fractional-Laplacian definitions in R^d yield distinct operators because their boundary conditions differ: spectral models use local boundary data, whereas Riesz models require exterior data.The stochastic interpretation attributes this distinction to the noncommuting order of stopping Brownian motion and subordinating it, and the paper uses walk-on-spheres data to solve the Riesz Poisson equation.Source: §1.1 · pp. 4–6
• The study surveys commonly used fractional Laplacian definitions and quantitatively compares selected state-of-the-art numerical methods, emphasizing nonzero boundary conditions, stochastic methods, and new discretizations.The selected methods are not intended to cover every available numerical approach, but to reflect the authors’ expertise and support practitioners choosing definitions and solvers.Source: §1.1 · pp. 4–5
• The paper introduces a radial-basis collocation method for Riesz problems with nonzero Dirichlet data, proves equivalence among inhomogeneous spectral definitions, and proposes nonharmonic lifting for spectral Poisson problems.The Riesz method uses a directional vector Grünwald–Letnikov discretization and extends to complex, high-dimensional, and non-isotropic Lévy settings.Source: §1.1 · p. 5

1.2. Motivating Example

• For f(x) = 1, Riesz solution maxima vary non-monotonically with α, whereas spectral solution maxima increase monotonically; changing interval length can switch both trajectories among decreasing, non-monotonic, and increasing behavior.The switching occurs at different α values for the two definitions because u_L(x) = L^αu_1(x/L) and u_1 depends on α.Source: §1.2 · pp. 6–7
• For f = 1, Riesz solutions exceed spectral solutions throughout the interval, while their difference develops sharp boundary layers and derivative singularities as α falls below 1 and approaches zero.This ordering follows the positivity-preserving property of the inverse Riesz operator minus the inverse spectral operator for zero Dirichlet conditions.Source: §1.2 · p. 7
• With smooth forcing f = sin(πx), Riesz solutions develop increasingly sharp boundary layers as α → 0, whereas spectral solutions remain analytic because the forcing is a spectral-Laplacian eigenfunction.The contrast suggests the Riesz definition may better model anomalous-diffusion data exhibiting boundary layers, while sufficiently singular forcing can also produce singular spectral solutions.Source: §1.2 · p. 7

Section Overview

• The reviewed Riesz, spectral, directional, and regional definitions are equivalent on R^d but can diverge on bounded domains through different boundary conditions and regularity properties.The section also connects these operators to stochastic processes, extension methods, well-posedness, and discretization strategies for later analysis and computation.Source: Section Overview · p. 11, §2.1 · p. 11

2.1. The Fractional Laplacian on Rd

• Together, the R^d characterizations provide the foundation for the article’s bounded-domain operators, where boundary conditions determine whether the formulations remain equivalent.The section’s theoretical review therefore establishes the definitions used later for fractional Poisson problems, stochastic algorithms, and numerical discretization.Source: §2.1 · p. 11, Section Overview · p. 11

2.1.1. Spectral/Fourier Definition

• The spectral/Fourier definition constructs (−∆)^α/2 through the spectral theorem and, on R^d, identifies it as a Fourier multiplier with symbol |ξ|^α.The Fourier-transform formulation does not directly extend to bounded domains, whereas spectral functional calculus remains applicable with appropriately chosen spaces, spectra, and boundary conditions.Source: §2.1.1 · pp. 11–12

2.1.2. Singular Integral Representation

• The singular-integral representation expresses the fractional Laplacian in real space using a principal-value integral whose difference u(x) − u(y) regularizes the singularity.For negative powers, the related Riesz potential is obtained from the decaying multiplier |ξ|^α and is well defined for sufficiently smooth, decaying functions.Source: §2.1.2 · p. 13

2.1.3. Via the standard Laplacian: Elliptic Extension, Heat Semigroup, & Balakrishnan Formula

• Three equivalent formulations use standard Laplacian problems in one higher dimension: elliptic extension, heat semigroup evolution, and the Balakrishnan resolvent formula.For the spectral fractional Laplacian with zero Dirichlet data, the extension becomes cylindrical over the domain, while heat-kernel formulas can also accommodate nonzero boundary conditions.Source: §2.1.3 · pp. 13–14

2.1.4. Relation to L´evy processes

• The fractional Laplacian is the generator, up to sign, of standard isotropic α-stable Lévy motion, whose heavy-tailed jumps model anomalous diffusion.For α < 2, the motion has infinite variance and spreads like t^1/α; its generator form yields the singular-integral representation, with a sign change across α = 1.Source: §2.1.4 · pp. 14–17
• The Lévy-process connection supports stochastic Feynman-Kac and Brosamler formulations for boundary-value problems involving fractional Laplacians in bounded domains.These interpretations motivate Monte Carlo solution methods and connect the operator’s generator form to later directional representations.Source: §2.1.4 · p. 16

2.1.5. Directional Representation

• The directional representation integrates Riemann-Liouville fractional directional derivatives and reproduces the Fourier symbol |ξ|^α for α ≠ 1.The formula excludes α = 1, while choosing a uniform spherical measure recovers the Riesz fractional Laplacian; arbitrary measures generate anisotropic Lévy-stable operators beyond this article’s scope.Source: §2.1.5 · pp. 17–18

2.2. Fractional Laplacians on Bounded Domains

• On bounded domains, the Riesz and directional fractional Laplacians require exterior data on R^d \ Ω, whereas the spectral fractional Laplacian admits local boundary conditions on ∂Ω.These distinctions arise because whole-space characterizations cease to be equivalent after restriction to bounded domains.Source: §2.2 · p. 18

2.3.1. Definition

• The Riesz fractional Laplacian depends on Ω and exterior values g, while its Neumann formulation remains unsettled because no widely accepted exterior fractional normal derivative exists.Nonzero exterior Dirichlet conditions have been treated with finite elements, Monte Carlo, radial basis collocation, and Lagrange multiplier methods.Source: §2.3.1 · p. 19, §2.3.2 · pp. 19–20

2.3.3. Stopped L´evy Motion

• With Dirichlet conditions, the Riesz fractional Laplacian generates stopped α-stable Lévy motion whose discontinuous paths typically exit Ω by jumping into R^d \ Ω.This connection yields a Feynman–Kac representation, and walk-on-spheres accelerates its computation using maximal inscribed spheres.Source: §2.3.3 · p. 20

2.3.4. Well-posedness and Regularity

• For bounded Lipschitz Ω, the Riesz Poisson problem is well-posed for f(x) ∈ H^-α/2(Ω), but its boundary regularity is limited relative to the spectral definition.Increasing source regularity may improve only interior regularity once α/2+s ≥ 1/2; for f ∈ L2(Ω), global H^α regularity holds spectrally for α ∈ (0, 2) but for Riesz only when α < 1.Source: §2.3.4 · pp. 21–22

2.4. Directional Fractional Laplacian

• Using the uniform spherical measure, the directional fractional Laplacian has the same stopped symmetric stable Lévy process as the Riesz fractional Laplacian, so their discussions of stochastic behavior and regularity coincide.Although the directional formulation can represent anisotropic anomalous diffusion, this work selects the uniform measure for comparison with Riesz.Source: §2.4 · p. 22

2.5.1. Zero Boundary Condition

• The homogeneous spectral fractional Laplacian is defined through discrete Laplacian eigenvalues and eigenfunctions, requires no exterior information, and supports the same local Dirichlet or Neumann boundary-condition types as −∆.Other representations include extension, heat-semigroup, Balakrishnan, and pseudo-parabolic formulations, but most homogeneous characterizations require modification for nonzero boundary data.Source: §2.5.1 · p. 23, §2.5.2 · p. 23

2.5.3. Subordinate Stopped/Reflected Brownian Motion

• The spectral fractional Laplacian generates subordinate stopped Brownian motion with Dirichlet conditions and subordinate reflected Brownian motion with Neumann conditions; imposing boundaries before subordination distinguishes it from Riesz.The reverse order—subordinating first and then stopping—corresponds to the Riesz fractional Laplacian in Ω.Source: §2.5.3 · p. 24

2.6. Inhomogeneous Spectral Fractional Laplacian

• For nonzero Dirichlet data, harmonic lifting reduces the inhomogeneous spectral fractional Laplacian to the homogeneous operator, and the constructions of Antil et al. [18] and Cusimano et al. are equivalent.The resulting operator does not approach the identity as α → 0 when g ≠ 0; the series representation extends the spectral expansion to smooth functions with nonzero boundary values.Source: §2.6 · pp. 24–25, §2.6.1 · p. 25

2.6.1. Series Representation

• The inhomogeneous spectral fractional Laplacian is defined using a single-valued inverse Laplacian fixed by boundary data, with the zero gauge yielding a spectral series representation.The zero-boundary inverse removes the inverse Laplacian’s harmonic ambiguity, while the resulting fractional inverse may act on functions with arbitrary boundary values.Source: §2.6.1 · pp. 25–26

2.6.2. Harmonic Lifting

• For sufficiently regular boundary data, the fractional harmonic lifting in Antil et al. [18] equals the solution of the standard Laplace equation and has an equivalent very weak variational formulation.When g ∈H1/2(∂Ω), the lifting satisfies b ∈H1(Ω) and can be treated in the weak sense.Source: §2.6.2 · p. 27

2.6.3. Nonharmonic Lifting

• A nonharmonic lifting v ∈H1(Ω) leads to a weak definition in which fractional operators act on zero-boundary functions, requiring g ∈H1/2(∂Ω).The solution is recovered from u = w + v, and the lifting need not be unique.Source: §2.6.3 · p. 28

2.6.4. Heat Semigroup

• The heat-semigroup formulation of Cusimano et al. is equivalent to Antil et al.’s [18] formulation because both apply the homogeneous operator to a harmonic-lifted function.The equivalence is also demonstrated numerically in Section 5.1.2.Source: §2.6.4 · p. 29

2.6.5. Well-posedness and Regularity

• For the spectral fractional Poisson problem, existence and uniqueness hold for general nonzero boundary conditions, with minimal data g ∈L2(∂Ω) for Dirichlet and g ∈H−1(∂Ω) for Neumann.If s < 1/2, f ∈Hs(Ω) implies u ∈Hs+α(Ω); if 1/2 < s < 2 + 1/2, only u ∈H1/2−ϵ+α(Ω) is guaranteed unless f = 0 on ∂Ω, and the α-gain is sharp. For Neumann problems, when 3/2 < s < 7/2, u ∈Hs+α(Ω) holds if and only if ∂f/∂n = 0 on ∂Ω.Source: §2.6.5 · pp. 30–31

• The Riesz Laplacian corresponds to processes that leave the domain’s closure, whereas the spectral Laplacian corresponds to processes confined to it.These distinctions are summarized alongside their fundamental properties in Table 2.Source: §2.7 · p. 32, Table 2 · p. 32
• The regional fractional Laplacian is another definition whose functions are defined on Ω rather than all of R^d, and it differs from the Riesz definition even for zero exterior data.Source: §2.7 · pp. 32–33
• The regional Laplacian’s Poisson problem has been studied through the Feynman–Kac formula, with reflected and censored α-stable processes discussed in.Neumann and Robin boundary conditions for the regional Laplacian were discussed in.Source: §2.7 · p. 33
• Other fractional Laplacians arise from related processes [93], within a broad probabilistic literature on stable-type processes in bounded domains.Probabilistic techniques have also been used to derive spectral-Laplacian eigenvalue estimates.Source: §2.7 · p. 33

Section Overview

• The study combines AFEM and Walk-on-Spheres for the Riesz operator, spectral element discretization for the spectral operator, a new RBF-vector Gr¨unwald method for the directional operator, and finite volumes for horizon-based models.The vector Gr¨unwald scheme is implemented numerically here for the first time to the authors’ knowledge.Source: Section Overview · p. 34

3.1.1. Adaptive Finite Element Method (AFEM)

• The Riesz formulation imposes a nonlocal exterior Dirichlet condition, while AFEM addresses singular integrals, dense systems, and boundary-induced low regularity through adaptive refinement.Residual error estimates and Dörfler marking produce meshes highly refined near the boundary and achieve optimal convergence rates with piecewise linear elements.Source: §3.1 · p. 34, §3.1.1 · p. 35
• For α<1, AFEM approximants may fail to enforce the zero boundary condition strongly because Hα/2(Ω) need not provide the required trace, although boundary oscillations diminish under refinement.The true weak solution can still possess a zero trace when α+s>1/2, while the approximation space may not enforce it strongly.Source: §3.1.1 · p. 35

3.1.2. Walk-on-Spheres (WOS) Method

• Walk-on-Spheres solves Riesz problems stochastically using killed isotropic α-stable processes, whose jumps permit disconnected domains and finite-step termination under stated domain conditions.The method is embarrassingly parallel and converges for convex domains and bounded non-convex domains satisfying a uniform exterior-cone condition.Source: §3.1.2 · pp. 35–37

3.2. Spectral Definition

• The spectral fractional Poisson problem can be reformulated as a weighted elliptic extension on a semi-infinite cylinder, enabling standard discretization methods; this work uses the spectral element method for accuracy and implementation ease.Alternative approaches include truncation-based finite elements, hybrid finite element-spectral discretization, heat semigroup methods, and elliptic extensions.Source: §3.2 · pp. 37–38

3.2.1. Discrete Eigenfunction and Spectral Element Methods

• The discrete eigenfunction and spectral element methods approximate the spectral operator through generalized eigenpairs of stiffness and mass matrices, but rigorous stability and accuracy proofs remain unpublished.The discrete eigenfunction method is used in one dimension, while spectral elements are used for higher-dimensional domains.Source: §3.2.1 · p. 39

3.3. Directional Representation

• For the uniform directional measure, the directional operator agrees with the Riesz operator; the paper introduces an RBF collocation method using a modified vector Gr¨unwald-Letnikov scheme and directly enforces the full exterior condition.The method focuses on the uniform case, while non-uniform directional measures are left for future work.Source: §3.3 · pp. 39–40, §3.3.1 · p. 40

3.3.1. A Radial Basis Function Collocation Method for the Riesz fractional Laplacian

• The modified vector Gr"unwald-Letnikov formula has first-order spatial accuracy, while its user-selected truncation parameter K2 controls truncation error.For compactly supported exterior data, K2h can exceed the support diameter; for decaying data, truncation must reflect the decay rate.Source: §3.3.1 · p. 41
• For the unit-disk test with g(x)=exp(−|x|2), K2=6000 makes truncation errors near machine precision, while the final RBF solution is already converged near K2=2000.At h=0.001, truncation is isotropic at the disk center but direction-dependent near the boundary.Source: §3.3.1 · p. 42, Figure 6 · p. 42, Figure 10 · p. 47

3.3.2. Validation Using Reference Solutions and Walk-on-Spheres

• The modified GL discretization achieves O(h) convergence, matching the theoretical rate given in.The validation uses two functions on the unit disk with zero exterior condition.Source: §3.3.2 · p. 44, Figure 7 · p. 45
• The RBF collocation method converges for zero exterior conditions, but accuracy decreases for sharper boundary gradients and errors are larger near the boundary.Increasing collocation points eventually reduces convergence rates because the collocation matrix becomes increasingly ill-conditioned.Source: §3.3.2 · p. 44, Figure 8 · p. 45, Figure 9 · p. 46
• For nonzero exterior data, both RBF and Walk-on-Spheres produce accurate solutions, while WOS error decays as O(NT^-0.5) and RBF convergence is inconsistent.At relative L2 error 10^-3, WOS takes about 900 seconds versus about 1100 seconds for RBF using 24 cores.Source: §3.3.2 · pp. 46–47, Figure 11 · p. 48

3.3.3. Future Directions

• Future work targets effective preconditioners for RBF matrices and theoretical error bounds for truncated directional fractional Laplacians.The broader goal is to address general directional fractional Laplacians connected to multivariate stable processes.Source: §3.3.3 · p. 47

3.4. Horizon-based Nonlocal Definition

• The horizon-based formulation restricts interactions to a finite domain and approaches the Riesz fractional Laplacian as the horizon tends to infinity.Its finite-volume discretization yields a symmetric, diagonally dominant Toeplitz matrix with a unique solution and O(N log N) solution cost using suitable preconditioned iteration.Source: §3.4 · p. 48, §3.4.1 · p. 49, §3.4.2 · p. 50

3.4.3. Numerical Examples

• The experiments vary δ = 2L and 64L, α = 0.5 and 1.5, and use the N = 215 numerical solution as a reference when no exact solution exists.A separate horizon study uses N = 214 with f(x) = 1 and f(x) = sin(πx).Source: §3.4.3 · pp. 50–51
• Finite-volume L2-error convergence is about O(h^1.2) for α = 0.5 and O(h^1.5) for α = 1.5, while L∞ rates are about O(h^0.5) and O(h^0.7), respectively.These rates are reported from Figure 12; theoretical finite-volume convergence estimates were unavailable, although finite-element L2 estimates exist [110].Source: §3.4.3 · p. 50, Figure 12 · p. 50
• For the Riesz fractional diffusion problem, the finite-volume integral is partitioned so the first term uses rectangle-rule discretization while the last two terms are computed exactly.The exterior-zero condition makes x + z lie outside the domain for sufficiently large |z|, enabling the exact treatment.Source: §3.4.3 · p. 51
• The L∞ convergence rate of ||u − uδ||∞ versus horizon δ is 0.5 for α = 0.5 and 1.5 for α = 1.5, with convergence to the fractional diffusion solution as δ → ∞.This behavior is illustrated in Figures 13–14 and was also observed by D’Elia and Gunzburger.Source: §3.4.3 · p. 51, Figure 14 · p. 52, Figure 13 · p. 52

Section Overview

• Four benchmark problems on square, disk, and L-shaped domains show operator-dependent solutions for α = 0.5 and α = 1.5, with directional and Riesz solutions equivalent up to numerical error.Multiple numerical methods were used to verify that observed differences reflect the operators rather than discretization inaccuracies.Source: Section Overview · p. 53, §4.1 · p. 54

4.1. Square Domain

• On the square, the Riesz solution lies above the spectral solution for f ≥ 0, while α = 0.5 produces boundary oscillations and sharper Riesz boundary layers than the spectrally smooth solution.The apparent failure of the Riesz approximation to enforce zero trace at α = 0.5 is an AFEM artifact; the true Riesz solution has zero trace.Source: §4.1 · pp. 53–54

4.2. Disk Domain

• On the disk, α = 0.5 again yields Riesz boundary oscillations, sharper Riesz boundary layers, and Riesz solutions above spectral solutions when f ≡ 1.The comparisons for sinusoidal forcing are similar to the square-domain cases, while directional comparisons are reported as similar in Appendix C.Source: §4.2 · p. 57

4.3. L-Shaped Domain

• On the L-shaped domain, difference plots show a relatively minor interior-corner spike that is intensified for α = 0.5 and much less pronounced for α = 1.5.For f ≥ 0, the Riesz solution again lies above the spectral solution, consistent with the theoretical result of.Source: §4.3 · p. 58

4.4. Computational Considerations

• AFEM is a robust Riesz discretization with complexity O(n(log n)2d) operations and accurately captures boundary singularities, but it has currently been applied only to zero Dirichlet conditions.Its complexity includes assembly, adaptive error indicators, and linear-system solution.Source: §4.4 · p. 62
• WOS supports quasi-convex domains and zero or nonzero Dirichlet data, scales with sample paths times domain points, and is embarrassingly parallel.Nonzero boundary data require only inserting g(x) into the Feynman–Kac formula (31).Source: §4.4 · p. 62
• RBF collocation handles non-uniform directional measures and decaying nonzero exterior data, but close collocation points cause instability and linear solves have cubic or squared complexity for direct or iterative solvers.Good preconditioners are unavailable because of the complicated matrix structure produced by the Grünwald–Letnikov scheme and Gaussian quadrature.Source: §4.4 · p. 62
• The spectral SEM accommodates varied boundary conditions and complex domains, but eigenpair computation and the resulting linear system are computationally expensive; reused eigenpairs make time-dependent solves inexpensive after initialization.The resulting linear system has cubic complexity, while time-dependent applications reuse the eigenpairs at every time step.Source: §4.4 · pp. 62–63

Section Overview

• Nonzero boundary conditions can substantially increase computational cost, require modified definitions for the spectral case, and motivate feasible Riesz Dirichlet solvers; fractional Neumann problems remain unresolved for the Riesz definition.Riesz Neumann problems lack consensus on the nonlocal condition and suitable numerical methods, whereas spectral Neumann conditions are local.Source: Section Overview · p. 64

5.1. Spectral Definition

• The Antil et al. [18], Cusimano et al., and nonharmonic lifting formulations numerically describe equivalent inhomogeneous spectral fractional Laplacians.These formulations require distinct numerical approaches, which are compared in the following subsections.Source: §5.1 · p. 64

5.1.1. Numerical Comparison of Nonharmonic Lifting and Harmonic Lifting Methods

• For f = x and f = −x, differences between APR harmonic lifting [18] and nonharmonic lifting are on the order of machine precision across different fractional orders α and lifting functions v = x and v = x3.Source: §5.1.1 · p. 65, Figure 24 · p. 65

5.1.2. Numerical Comparison of the Harmonic Lifting and Heat Semigroup Approaches

• The harmonic lifting and heat semigroup approximations use the same spectral-element spatial discretization but differ in time integration, quadrature, truncation, and computational setup.The comparison evaluates α = 0.5 and α = 0.8 with N = 20, time step 6e-5, 200,000 Euler iterations, 30 quadrature points, and T = 1.Source: §5.1.2 · pp. 65–66
• For time-independent fractional equations, the heat semigroup formulation may be faster, whereas APR with spectral elements [18] is expected to be more efficient for time-dependent problems.Heat semigroup discretization repeatedly solves a heat equation over many time steps; APR computes N eigenvalues and eigenfunctions per element with complexity N3, then reuses the eigenpairs across time steps.Source: §5.1.2 · p. 66, §5.2 · p. 67

5.2. Comparison of Spectral, Directional, and Riesz Solutions

• With nonzero Dirichlet data, directional and Riesz solutions remain equivalent up to numerical error, while the spectral solution has greater magnitude, reversing the zero-boundary observation.The example uses α = 1.5, with boundary data posed locally for the spectral definition and on the exterior for the Riesz and directional definitions.Source: §5.2 · pp. 66–67, Figure 27 · p. 68
• For the RBF collocation solution with g(x) = exp(−|x|2), convergence is reached at truncation parameter K2 = 6000 because the exterior condition decays rapidly.Table 4 varies K2 from 1000 to 6000 for the unit-square inhomogeneous fractional Poisson problem with f(x) = 1.Source: §5.2 · p. 67, Table 4 · p. 69

Methods and theory

• The work develops and compares numerical methods, presents a new radial basis function collocation method, and analyzes spectral, horizon-based, and Riesz formulations on one- and two-dimensional benchmarks.It also surveys implementation details, regularity, well-posedness, bounded-domain restrictions, and stochastic connections.Source: §6 · p. 70

Inhomogeneous problems

• All considered approaches for inhomogeneous fractional Laplacians produced equivalent benchmark solutions, an equivalence proved analytically for the first time.Nonlocality creates computational challenges even with zero Dirichlet data, while nonzero boundary conditions further increase difficulty.Source: §6 · p. 70

Scope and limitations

• The numerical-method discussion is intentionally selective rather than exhaustive, omitting extended treatment of finite-difference approaches despite emphasizing state-of-the-art methods relevant to method selection.The authors frame the work as focused on fundamental questions, especially nonzero boundary conditions.Source: §6 · p. 70

Numerical observations

• Riesz fractional Poisson solutions develop boundary singularities while spectral solutions remain smooth for nonsingular source functions, making boundary regularity important when selecting a definition.The study also found that domain size significantly affects numerical solution evolution as the fractional order α changes.Source: §6 · p. 70

Future directions

• The study provides a starting point for researchers applying fractional Laplacians to systems exhibiting anomalous transport, while noting that the subject remains incomplete.The work identifies directions for future development and practical choices among computational approaches.Source: §6 · p. 70

• The appendix collects fractional Sobolev spaces used for the fractional Laplacian, following the exposition and notation of [18] while also referencing.Source: Appendix A · p. 81
• For 0 < s < 1, it defines the fractional Sobolev space together with its seminorm and norm.Source: Appendix A · p. 81
• For smooth functions on Ω, the appendix defines a trace operator γ.Source: Appendix A · p. 81
• On bounded, simply connected Lipschitz domains, the trace theorem applies for 1/2 < s < 3/2 and provides the stated trace result [116].The appendix notes that this range is sufficient for its discussion of traces.Source: Appendix A · p. 81
• For s > 1/2, it defines a corresponding subspace, while s = 1/2 requires a separately defined fractional Sobolev space.Source: Appendix A · p. 81

The associated norm is defined

• Hs(Ω) is defined to characterize regularity properties of the spectral fractional Laplacian.Source: The associated norm is defined · p. 81
• The spectral characterization uses eigenpairs (λk, φk) of the integer Laplacian −∆ on Ω with zero Dirichlet boundary conditions.Source: The associated norm is defined · p. 81
• A relationship between the fractional Sobolev spaces in Definitions A.1, A.3, A.4, and A.5 is noted in Ref. [18].Source: The associated norm is defined · p. 82
• For s ≥ 0, H−s(Ω) is defined as the dual space of Hs(Ω).Source: The associated norm is defined · p. 82

Appendix B Grids

• The appendix provides the meshes and collocation points used for the two-dimensional numerical comparisons in Section 4.The figures cover the square, disk, and L-shaped domain, with method-specific discretizations for directional, spectral, and Riesz fractional Poisson equations.Source: Appendix B · p. 83, Figure 31 · p. 83, Figure 32 · p. 84

Additional Disk Comparisons

• For α = 0.5 and f = sin(πr^2), the disk-domain spectral and Riesz solutions are shown together with their pointwise difference.The spectral solution is computed using SEM and the Riesz solution using AFEM.Source: Appendix C · p. 86
• Figure 34 presents disk-domain solutions and the difference between u_Riesz and u_spectral for α = 1.5.Source: Figure 34 · p. 86
• For α = 1.5 with f = 1 and g(x) = 0, the spectral solution differs significantly from the Riesz solutions, which agree up to numerical error across three computational methods.The Riesz solutions use directional RBF collocation, AFEM, and WOS, while the spectral solution uses SEM.Source: Figure 35 · p. 86

Additional L-shape Comparisons

• All three Riesz solutions are equivalent up to numerical error, whereas the spectral solution shows the only significant difference.The comparison uses RBF collocation with the directional representation, AFEM, and WOS for the Riesz solution, and SEM for the spectral solution.Source: Figure 36 · p. 87
• The spectral solution is computed with SEM, while the Riesz solutions are computed using RBF collocation, AFEM, and WOS.Source: Figure 36 · p. 87
• A view facing the outside corner presents the same comparison as Figure 36.Source: Figure 37 · p. 87