Stability of Galerkin and Inertial Algorithms with variable time step size

This paper provides Galerkin and Inertial Algorithms for solving a class of nonlinear evolution equations. Spatial discretization can be performed by either spectral or finite element methods; time discretization is done by Euler explicit or Euler semi-implicit difference schemes with variable time step size. Moreover, the boundedness and stability of these algorithms are studied. By comparison, we find that the boundedness and stability of Inertial Algorithm are superior to the ones of Galerkin Algorithm in the case of explicit scheme and the boundedness and stability of two algorithms are same in the case of semi-implicit scheme.