Spectral fractional Laplacian

SpectralFractionalLaplacian(u, s, bcs=...) is the fractional power of the homogeneous-Dirichlet discrete Laplacian. With eigenpairs \(\{(\lambda_k,q_k)\}\) of the positive Dirichlet Laplacian, orthonormal in \(L^2(\Omega)\),

\[ (-\Delta_D)^s u=\sum_k \lambda_k^s (u,q_k)_{L^2(\Omega)}q_k . \]

The finite-element eigenproblem is \(Kq_k=\lambda_kMq_k\) with Dirichlet constraints and \(q_i^TMq_j=\delta_{ij}\), so the discrete operator is

\[ A_h^su=Q\,\operatorname{diag}(\lambda_k^s)Q^TMu, \qquad A_h=M^{-1}K . \]

Discretization

Yonderdrake never forms the eigendecomposition. It evaluates the Balakrishnan integral representation of \(A_h^s\) with sinc quadrature, which turns each application into a sum of shifted elliptic solves that stay distributed.

  • sinc_truncation_target controls the quadrature model only. It is a truncation estimate for the sinc rule, not a bound on the finite-element error. Targets below meaningful float64 precision are clamped with a warning. Requests requiring more than 100,000 nodes are rejected.

  • shift_cache trades setup cost against memory: "stream" keeps two shifted solvers, "all" caches one matrix and KSP per shift.

  • diagnostics() reports node count, model estimate, setups, assemblies, solves, and reuse.

Requirements: \(0<s<1\) and complete homogeneous exterior Dirichlet conditions. Other boundary realizations are unsupported.

Refine the sinc target on a fixed mesh before measuring finite-element convergence. On a coarse mesh the discretization error dominates and tighter sinc targets change nothing.

Sources

Bonito and Pasciak (2015) for fractional powers of elliptic operators, and Bonito, Lei, and Pasciak (2019) for the sinc quadrature analysis.

This is not the zero-exterior Riesz operator or the periodic Fourier operator. See the three spatial operators are not interchangeable.