Before revision:  Assuming this restriction, an exponential rate of convergence of the PAGD sequence is demonstrated, by mimicking the convergence of the solution to the ODE via energy methods. Application of the PAGD method is made in the context of solving certain nonlinear elliptic PDE using pseudo-spectral methods  , and several numerical experiments are conducted.
After revision:  Assuming this restriction, an exponential rate of convergence of the PAGD sequence is demonstrated, by mimicking the energy argument and the convergence at the continuous level. Applications of the PAGD method is made in the context of solving certain nonlinear elliptic PDE using pseudo-spectral methods , and several numerical experiments are conducted.