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

• Published in: International journal of heat and mass transfer Vol. 120; pp. 1132 - 1145
• Main Authors: Liu, Yang, Yu, Zudeng, Li, Hong, Liu, Fawang, Wang, Jinfeng
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.05.2018; Elsevier BV
• Subjects: A priori error estimates; Algorithms; Coarsening; CPU time; Error analysis; Finite element analysis; Finite element method; Heat transfer; Interpolation; Iterative methods; Mathematical models; Stability; Time fractional water wave model; Time two-mesh finite element algorithm; Stability; CPU time; Time two-mesh finite element algorithm; Time fractional water wave model; A priori error estimates
• ISSN: 0017-9310; 1879-2189
• DOI: 10.1016/j.ijheatmasstransfer.2017.12.118

•The new TT-M algorithm combined with finite element method is proposed.•The TT-M method can save the CPU-time.•The TT-M method have some applications in solving nonlinear PDEs.•The TT-M method can be combined with other numerical methods. 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.