A Simple Solver for the Fractional Laplacian in Multiple Dimensions
By Victor Minden, Lexing Ying. 2018
66 citations recorded by the source corpus.
We present a simple solver of the fractional Laplacian based on the hypersingular integral representation. Through singularity subtraction, we obtain a regularized integrand that is amenable to the trapezoidal rule with equispaced nodes. The resulting quadrature scheme gives a discrete operator on a regular grid that is translation-invariant and thus can be applied quickly with the fast Fourier transform. For discretizations of problems related to space-fractional diffusion on bounded domains, we observe that the underlying linear system can be efficiently solved via preconditioned Krylov methods with a preconditioner based on the finite-difference (non-fractional) Laplacian. We show numerical results illustrating the error of our simple scheme as well the efficiency of our preconditioning approach, both for the elliptic (steady-state) fractional diffusion problem and the time-dependent problem.
Subfields: Atomic and Molecular Physics, and Optics, Mechanics of Materials, Modeling and Simulation
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In Atomic and Molecular Physics, and Optics
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Citations in the corpus
- fPINNs: Fractional Physics-Informed Neural Networks (2018, methodology): “ensomeworksonsolving3Dspace-fractionalADEs. In [36,37,38,39],onlythe3Dextension oftheRieszspace-fractionalderivativesisconsidered,whichdiffersfromthedirectionalfractional Laplacian of our interest; in [40], Riesz fractional Laplacian is first regularized by singularity subtraction and then approximate”