Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
For researchers in fractional PDEs, this is an incremental survey that organizes known results but does not present new methods or breakthroughs.
This paper provides a concise overview of numerical methods for time-fractional evolution equations (subdiffusion model) with nonsmooth data, covering regularity theory, Galerkin FEM, time-stepping schemes, and space-time variational formulations, with comparisons to standard parabolic problems. It includes illustrative numerical experiments and literature pointers.
Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $α\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.