Time two-mesh algorithm combined with finite element method for time fractional water wave model

In this article, a new time two-mesh (TT-M) finite element (FE) method, which is constructed by a new TT-M algorithm and FE method in space, is proposed and analyzed. The numerical theories and algorithm are shown by solving the fractional water wave model including fractional derivative in time. The TT-M FE algorithm mainly covers three steps: firstly, a nonlinear FE system at some time points based on the time coarse mesh ΔtC is solved by an iterative method; further, based on the obtained numerical solution on time coarse mesh ΔtC in the first step, some useful numerical solutions between two time coarse mesh points are arrived at by the Lagrange's interpolation formula; finally, the solutions on the first and second steps are chosen as the initial iteration value, then a linear FE system on time fine mesh ΔtF<ΔtC is solved. Some stable results and a priori error estimates are analyzed in detail. Furthermore, some numerical results are provided to verify the effectiveness of TT-M FE method. By the comparison with the standard FE method, it is easy to see that the CPU time can be saved by our TT-M FE method.