Numerical Methods for Fractional Diffusion

Abstract

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford-Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.

References in corpus (17)

• An extension problem related to the fractional Laplacian
• The scaling laws of human travel
• Fractional Quantum Mechanics and Levy Path Integrals
• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• A fractional Laplace equation: regularity of solutions and Finite Element approximations
• Anomalous Diffusion in View of Einstein's 1905 Theory of Brownian Motion
• A short FE implementation for a 2d homogeneous Dirichlet problem of a Fractional Laplacian
• Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains
• Regularity theory and high order numerical methods for the (1d)-Fractional Laplacian
• Tensor FEM for spectral fractional diffusion
• Eigenvalues of the fractional Laplace operator in the unit ball
• Spectral results for mixed problems and fractional elliptic operators
• A PDE approach to fractional diffusion: a posteriori error analysis
• $hp$-Finite Elements for Fractional Diffusion
• Finite element approximations of the nonhomogeneous fractional Dirichlet problem
• Convergence rates for the classical, thin and fractional elliptic obstacle problems

Cited by in corpus (27)

• Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
• Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications
• Tensor FEM for spectral fractional diffusion
• A priori error estimates for the optimal control of the integral fractional Laplacian
• Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
• Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian
• Parameter-robust methods for the Biot-Stokes interfacial coupling without Lagrange multipliers
• Local convergence of the FEM for the integral fractional Laplacian
• A fractional PDE model for turbulent velocity fields near solid walls
• Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm
• Numerical continuation for fractional PDEs: sharp teeth and bloated snakes
• Weighted analytic regularity for the integral fractional Laplacian in polygons
• Nonlocal minimal graphs in the plane are generically sticky
• Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons
• Numerical algorithm for the space-time fractional Fokker-Planck system with two internal states
• Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons
• Finite elements for Matérn-type random fields: Uncertainty in computational mechanics and design optimization
• Plain convergence of adaptive algorithms without exploiting reliability and efficiency
• A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator
• Electroconvection of Thin Liquid Crystals: Model Reduction and Numerical Simulations
• Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian
• Lévy Flights and Leaky Boxes: Anomalous Diffusion of Cosmic Rays
• An a posteriori error estimator for the spectral fractional power of the Laplacian
• Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D
• A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains
• On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion
• An exponentially convergent discretization for space-time fractional parabolic equations using $hp$-FEM