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 is the weak action

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

not a nodal evaluation of a pointwise formula. The default edge-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, \(0<s<1\), 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.

Exact triangle-supported source action

Yonderdrake removes the source singularity analytically before quadrature. For a CG1 or CG2 polynomial on one affine triangle, extended by zero, the divergence theorem reduces the integral to edge integrals. For affine \(p\),

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

with \(d_e=(y-x)\cdot n_e\). Quadratic terms reduce to the same family of one-dimensional edge integrals.

This boundary reduction is a Yonderdrake derivation from the cited integral operator, verified against independently coded direct volume and polar integrals.

Backends

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

hmatrix compresses admissible far-field blocks with adaptive cross approximation, following Bebendorf (2000). compression_tolerance, admissibility, and leaf_size control it.

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