Conventions and normalizations¶

These definitions are normative for the whole library.

Notation¶

\(\alpha\) is a fractional time order, \(0<\alpha<1\).
\(s\) is a fractional power of the Laplacian, \(0<s<1\).
\(d\) is the spatial dimension.
\((-\Delta)^s\) is always written together with its realization, because the Dirichlet spectral, zero-exterior Riesz/restricted, and periodic Fourier operators are different operators.

Orders are immutable and must lie in the open unit interval.

Boundary terminology¶

Spectral: a fractional power of the homogeneous-Dirichlet or natural-Neumann elliptic operator. See Spectral fractional Laplacian.
Zero-exterior Riesz/restricted: a whole-space integral applied after extending the interior function by zero. See Riesz/restricted fractional Laplacian.
Periodic Fourier: a Fourier-series multiplier on a uniform flat periodic cell. See Periodic Fourier fractional Laplacian.

Fourier transform¶

For integrable \(u:\mathbb R^d\to\mathbb C\),

\[ \widehat u(\xi)=\int_{\mathbb R^d}e^{-i x\cdot\xi}u(x)\,dx, \qquad u(x)=\frac{1}{(2\pi)^d} \int_{\mathbb R^d}e^{i x\cdot\xi}\widehat u(\xi)\,d\xi , \]

so the whole-space fractional Laplacian has multiplier

\[ \widehat{(-\Delta)^s u}(\xi)=|\xi|^{2s}\widehat u(\xi). \]

On a periodic box of lengths \(L_j\), the continuous frequencies become \(2\pi k_j/L_j\). PeriodicFractionalLaplacian applies the corresponding discrete Fourier multiplier to the unique periodic nodal values and assigns zero to the constant mode.

Riesz normalization¶

Consistency with that multiplier fixes

\[ C_{d,s} =\frac{4^s\Gamma(d/2+s)} {\pi^{d/2}|\Gamma(-s)|} =\frac{2^{2s}s\Gamma(d/2+s)} {\pi^{d/2}\Gamma(1-s)}, \qquad C_{2,s}=\frac{2^{2s}s\Gamma(1+s)}{\pi\Gamma(1-s)} . \]

All Riesz backends share this constant.

Finite-element action¶

The Riesz implementation assembles the Galerkin action against every finite-element test function \(v\),

\[ \langle v,Lu\rangle=\int_\Omega v(x)Lu(x)\,dx , \]

The dense, matrix-free, and H-matrix backends all compute this action. Since the continuous operator is symmetric, the discrete identity \(\langle v,Lu\rangle=\langle u,Lv\rangle\) is used to check the implementation. The assembled action is left unchanged, preserving discrete asymmetry as a diagnostic.

Numerical precision and requested accuracy¶

The default methods compute quadrature rates and weights with standard 64-bit floating-point arithmetic (float64). Higher-precision values are used only for validation.

For the default spectral sinc quadrature, a target below meaningful float64 precision is raised to machine epsilon with a warning. A target requiring more than 100,000 sinc nodes raises an error.