A novel two-grid Crank-Nicolson mixed finite element method for nonlinear fourth-order sin-Gordon equation

A new nonlinear fourth-order sin-Gordon equation with actual physical background is first created. Then, by introducing an auxiliary function, the nonlinear fourth-order sin-Gordon equation is decomposed into the nonlinear system of equations of second-order derivatives of spatial variables. Subsequently, the time derivative is discretized by using the Crank-Nicolson (CN) scheme to construct a new time semi-discretized mixed CN (TSDMCN) scheme. Thereafter, the spatial variables in the TSDMCN scheme are discretized by using a two-grid mixed finite element (MFE) method to construct a new two-grid CN MFE (TGCNMFE) method with unconditional stability and second-order time precision, which consists of a system of nonlinear MFE equations defined on coarser grids and a system of linear MFE equations defined on finer grids with sufficiently high precision, and is very easy to solve. The existence, stability, and error estimates of the TSDMCN and TGCNMFE solutions are strictly proved theoretically, and the superiorities of the TGCNMFE method and the correctness of theoretical results are verified by two sets of numerical experiments.