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. target_quadrature_degree controls it, and target_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¶

For a CG1 or CG2 polynomial on one affine simplex, extended by zero, the endpoint route removes the source singularity analytically. 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 as their remaining edge integrals have Appell \(F_1\) primitives. 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 \(C_{3,s}\).

source_evaluation controls how each simplex source is evaluated:

Mode	Source action	Use
`hybrid` (default)	automatically choose source quadrature, edge quadrature, or an analytic endpoint formula	general use
`endpoint`	analytic boundary formula for every source and target pair	reference

source_quadrature_degree controls the Gauss rule for hybrid and has no numerical effect under endpoint. The admissibility parameter determines the near and far split. At a high level, hybrid uses:

2D
├─ well-separated → source-simplex quadrature
└─ near/coincident → analytic edge formula

3D
├─ well-separated → source-simplex quadrature
├─ moderately separated → boundary reduction + edge quadrature
└─ poorly separated → boundary reduction + analytic Appell formula

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.

Source quadrature, target quadrature, and compression have independent error controls. diagnostics() reports the selected source mode, source and target degrees, evaluation counts, storage, timings, solves, blocks, ranks, and achieved compression.