Numerical approximations for the variable coefficient fractional diffusion equations with non-smooth data
This work provides a rigorous numerical framework for solving fractional diffusion equations with variable coefficients, which is important for modeling anomalous diffusion in heterogeneous media.
The authors develop a spectral method for variable-coefficient fractional diffusion equations by transforming them to constant-coefficient equations, achieving error estimates that match numerical experiments for both smooth and non-smooth data.
In this article we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient fractional diffusion equation of the same order. The transformed equation retains the desirable stability property of being an elliptic equation. A spectral approximation scheme is proposed and analyzed for the transformed equation, with error estimates for the approximated solution derived. An approximation to the unknown of the variable coefficient fractional diffusion equation is then obtained by post processing the computed approximation to the transformed equation. Error estimates are also presented for the approximation to the unknown of the variable coefficient equation with both smooth and non-smooth diffusivity coefficient and right-hand side. Three numerical experiments are given whose convergence results are in strong agreement with the theoretically derived estimates.