A new two-grid mixed finite element Crank-Nicolson method for the temporal fractional fourth-order sine-Gordon equation

A new nonlinear temporal fractional fourth-order sine-Gordon (NTFFOSG) equation with practical physical significance is first developed. Then, by introducing an auxiliary function, the NTFFOSG equation is decomposed into the nonlinear system of equations with the second-order derivatives in spatial variables. Subsequently, by using the Crank-Nicolson (CN) scheme to discretize time derivative and time fractional derivative, a new time semi-discretization mixed CN (TSDMCN) scheme is constructed. Finally, by using two-grid mixed finite element (MFE) method to discretize the spatial variables in the TSDMCN scheme, a new two-grid MFE CN (TGMFECN) method with unconditional stability is established, which consists of a system of nonlinear MFE equations on coarser grids and a system of linear MFE equations on fine grids with sufficient precision, so it is very easy to solve. The largest contribution of this article is to theoretically analyze the existence, stability, and error estimates of the TSDMCN and TGMFECN solutions, and to verify the correctness of theoretical results and the superiority of the TGMFECN method through numerical experiments.