# Numerical approximations for the variable coefficient fractional diffusion equations with non-smooth data

## TL;DR

This paper develops a spectral approximation scheme for variable coefficient fractional diffusion equations, transforming them into constant coefficient forms to analyze stability and error, with validation through numerical experiments.

## Contribution

It introduces a novel spectral method for variable coefficient fractional diffusion equations by transforming them into constant coefficient problems, providing error estimates for smooth and non-smooth data.

## Key findings

- Spectral scheme achieves optimal convergence rates.
- Error estimates are valid for non-smooth coefficients.
- Numerical experiments confirm theoretical predictions.

## Abstract

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.