Finite element methods for fractional diffusion equations

Abstract

Due to the successful applications in engineering, physics, biology, finance, etc., there has been substantial interest in fractional diffusion equations over the past few decades, and literatures on developing and analyzing efficient and accurate numerical methods for reliably simulating such equations are vast and fast growing. This paper gives a concise overview on finite element methods for these equations, which are divided into time fractional, space fractional and time-space fractional diffusion equations. Besides, we also involve some relevant topics on the regularity theory, the well-posedness, and the fast algorithm.