An optimal order $$H^{1}$$ -Galerkin mixed finite element method for good Boussinesq equation

In this paper, by introducing an intermediate function, a splitting technique is employed for the fourth order time dependent non-linear Good Boussinesq equation. Then, an \(H^{1}\)-Galerkin mixed finite element method is applied to the Good Boussinesq (GB) equation with cubic spline space as test and trial space in the method. This method may be considered as a Petrov-Galerkin method in which cubic splines are trial and linear splines (i.e second derivative of cubic splines)as test space. Optimal order error estimates are obtained for the both semi discrete scheme and fully discrete scheme. The Numerical illustration is presented to support the theoretical analysis.