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In this note we consider the continuous Galerkin time‐stepping method of arbitrary order as a possible discretization scheme of nonlinear initial value problems. In addition, we develop and generalize a well‐known existing result for the discrete solution by applying a general linearizing procedure to the nonlinear discrete scheme including also the  simplified  Newton solution procedure. In particular, the presented existence results are implied by choosing sufficient small time steps locally. Furthermore, the established existence results are independent of the local approximation order. Moreover, we will see that the proposed solution scheme is able to significantly reduce the number of iterations. Finally, based on existing and well‐known a priori error estimates for the discrete solution, we present some numerical experiments that highlight the proposed results of this note.

Linearized continuous Galerkin h p ‐FEM applied to nonlinear initial value problems