Fractional space

All three spatial operators are external UFL operators. Build one and use it as an ordinary term.

Choosing a spatial realization

SpectralFractionalLaplacian raises the eigenvalues of a chosen Dirichlet operator,

\[ (-\Delta_D)^su=\sum_k\lambda_k^s(u,q_k)q_k, \]

while RieszFractionalLaplacian applies the whole-space singular integral to the zero extension,

\[ C_{2,s}\operatorname{PV}\int_{\mathbb R^2} \frac{u(x)-u(y)}{|x-y|^{2+2s}}\,dy . \]

PeriodicFractionalLaplacian acts on a flat periodic cell through the Fourier-series multiplier

\[ \widehat{(-\Delta_{\mathrm{per}})^s u}_k=|k_L|^{2s}\widehat u_k, \qquad k_L=2\pi(k_1/L_1,\ldots,k_d/L_d). \]

The three realizations have different domains, eigenfunctions, boundary behaviour, numerical parameters, and solvers. Dirichlet and zero-exterior realizations remain distinct under a homogeneous trace. Periodic tiling defines a third problem.

The same field on the same square produces three different responses. The Dirichlet spectral and zero-exterior Riesz realizations are shown on triangular meshes. The periodic Fourier realization is shown on the uniform quadrilateral periodic mesh required by its discrete Fourier transform.

The maze gallery experiment gives another comparison of the classical, spectral, and restricted Riesz operators on the same domain.

Spectral, Dirichlet or Neumann

Pass the homogeneous boundary condition explicitly:

Lu = SpectralFractionalLaplacian(
    u,
    0.4,
    bcs=bc,
    sinc_truncation_target=1.0e-6,
    shift_cache="all",
)
F = inner(CaputoDerivative(u, alpha), v) * dx + inner(Lu, v) * dx

Omit bcs for natural homogeneous Neumann conditions. Constants then form the zero eigenspace. A steady problem needs compatible forcing and a fixed mean, as in the maze gallery experiment.

The Koch snowflake and maze gallery experiments show the spectral operator evolving fields on nonrectangular domains.

Tighten sinc_truncation_target on a fixed mesh before measuring finite-element convergence, and inspect Lu.diagnostics() for node count and shifted-solver reuse.

Use shift_cache="all" for repeated small problems and "stream" for bounded storage.

Riesz, zero exterior

For \(s<1/2\) no boundary trace is required:

Lu = RieszFractionalLaplacian(u, 0.3)

This uses quadrature_degree=6, the boundary-adapted rule, and the matrix-free backend. The function is extended by zero outside the polygonal or polyhedral domain.

The supported spaces are scalar CG1 and CG2 on affine triangles in 2D or tetrahedra in 3D. Periodic or overlapping geometries are rejected. For \(s\geq1/2\), pass complete homogeneous bcs. A nonzero trace has infinite zero-extension energy.

A three-dimensional problem uses the same interface:

mesh = UnitCubeMesh(12, 12, 12)
V = FunctionSpace(mesh, "CG", 2)
x, y, z = SpatialCoordinate(mesh)
u = Function(V).interpolate(x * (1 - x) * y * (1 - y) * z * (1 - z))
bc = DirichletBC(V, 0.0, "on_boundary")
Lu = RieszFractionalLaplacian(
    u, 0.7, bcs=bc, assembly="hmatrix",
)

The integral geometry is substantially more expensive in 3D. The H-matrix backend is intended for larger meshes because it compresses interactions between well-separated basis-function supports. Refine compression_tolerance separately from quadrature_degree.

The Koch snowflake and maze gallery experiments compare this zero-exterior realization with the classical and spectral operators.

For larger problems, switch the backend and refine compression separately from quadrature:

Lu = RieszFractionalLaplacian(
    u, 0.3, bcs=bc, assembly="hmatrix", compression_tolerance=1e-8,
)

Backend trade-offs are tabulated in Riesz/restricted fractional Laplacian.

Periodic Fourier

Use a fully periodic uniform interval, quadrilateral rectangle, or hexahedral box:

mesh = PeriodicRectangleMesh(
    32, 24, 2*pi, 2*pi, quadrilateral=True, reorder=False,
)
V = FunctionSpace(mesh, "Q", 1)
u = Function(V)
Lu = PeriodicFractionalLaplacian(u, 0.4)

The three-dimensional construction uses the same operator:

mesh = PeriodicBoxMesh(
    24, 20, 16, 2*pi, 2*pi, 2*pi,
    hexahedral=True, reorder=False,
)
V = FunctionSpace(mesh, "Q", 1)
Lu = PeriodicFractionalLaplacian(Function(V), 0.62)

There are no boundary conditions or quadrature controls. Construction checks that the field has exactly one scalar degree of freedom at each periodic grid point and rejects a mesh that is nonuniform, only periodic in one direction, triangular, deformed, or higher order.

The periodic gyroid gallery experiment shows the three-dimensional operator acting on a multiscale periodic field. See Periodic Fourier fractional Laplacian for the exact scope.