Riesz/restricted fractional Laplacian¶

RieszFractionalLaplacian(u, s) is the whole-space singular integral applied to the zero extension of the interior function:

\[ (-\Delta)^s u(x) =C_{d,s}\,\operatorname{PV}\int_{\mathbb R^d} \frac{u(x)-u(y)}{|x-y|^{d+2s}}\,dy, \qquad u=0\quad\text{on }\Omega^c . \]

“Riesz” and “restricted” both refer to this construction, with the normalization \(C_{d,s}\) fixed in Conventions and normalizations so that the whole-space multiplier is \(|\xi|^{2s}\). The operator follows Di Nezza, Palatucci, and Valdinoci (2012) and its finite-element treatment follows Acosta and Borthagaray (2017).

Galerkin action¶

The assembled operator uses the weak Galerkin action

\[ A_{ij}=\int_\Omega\phi_i(x)(-\Delta)^s\phi_j(x)\,dx , \]

The default boundary-sector Gauss-Jacobi target rule resolves the remaining boundary singularity. quadrature_degree controls it, and quadrature_rule="ordinary" selects a Duffy tensor-Gauss alternative.

Scope: scalar CG1 or CG2 on affine 2D triangles or 3D tetrahedra, \(0<s<1\), zero exterior extension. Complete homogeneous bcs are required for \(s\geq1/2\), because a nonzero trace has infinite zero-extension energy. Below \(1/2\) no trace is needed. Periodic and overlapping cell geometries are rejected.

Simplex-supported source action¶

Yonderdrake removes the source singularity analytically before quadrature. For a CG1 or CG2 polynomial on one affine simplex, extended by zero, the divergence theorem reduces the volume integral to its boundary. Triangle sources reduce to analytic edge integrals. Tetrahedral sources reduce to triangular-face integrals whose radial direction is integrated analytically and whose tangential direction uses Gaussian quadrature. For an affine triangle polynomial \(p\),

\[ (-\Delta)^s(p1_T)(x) =\frac{C_{2,s}}{2s}\sum_{e\subset\partial T} \left[ p(x)d_e\int_e|y-x|^{-2-2s}\,dl +(\nabla p\cdot n_e)\int_e|y-x|^{-2s}\,dl \right], \]

with \(d_e=(y-x)\cdot n_e\). Quadratic terms reduce to the same family of one-dimensional edge integrals. In 3D the corresponding formula uses all four tetrahedron faces and the normalization \(C_{3,s}\).

Backends¶

Assembly	Storage	Parallel	Use
`matfree` (default)	no \(N^2\) matrix, replicated source geometry	MPI	general use, \(O(N^2)\) work
`hmatrix`	dense near blocks, low-rank far factors	serial or MPI	larger problems
`dense`	\(N^2\) weak entries	serial	reference

hmatrix compresses admissible far-field blocks with adaptive cross approximation, following Bebendorf (2000). compression_tolerance, admissibility, and leaf_size control it. This is particularly useful in 3D, where dense storage and uncompressed pairwise work become expensive quickly.

Quadrature error and compression error are separate and must be refined separately. diagnostics() reports storage, timings, solves, blocks, ranks, and achieved compression.