Finite difference/finite element method for a novel 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains

In this work, a novel two-dimensional (2D) multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on convex domains will be considered. Different from the general multi-term time-fractional diffusion-wave or sub-diffusion equation, the new equation not only possesses the diffusion-wave and sub-diffusion terms simultaneously but also has a special time-space coupled derivative. Although the analytic solution of this kind of equation can be derived using a multi-level sum of an infinite series and Fox H-functions, it is extremely complex and difficult to evaluate. Therefore, seeking the numerical solution of the equation is of great importance. In this paper, we will consider the finite element method for the novel 2D multi-term time fractional mixed diffusion equation. Firstly, we utilise the mixed L schemes to approximate the time fractional sub-diffusion term, diffusion-wave term and the coupled time-space derivative, respectively. Secondly, we establish the variational formulation and use the finite element method to discretise the equation. Then we adopt linear polynomial basis functions on triangular elements to derive the matrix form of the numerical scheme. Furthermore, we present the stability and convergence analysis of the numerical scheme. To show the effectiveness of our method, three examples are investigated, in which a 2D multi-term time fractional mixed diffusion equation on a circular domain and a 2D generalized Oldroyd-B fluid in a magnetic field are analysed.