Linearized Continuous Galerkin hp-FEM Applied to Nonlinear Initial Value Problems

TL;DR

This paper explores a high-order continuous Galerkin method for nonlinear initial value problems, providing existence results, efficiency improvements, and numerical validation for the proposed discretization scheme.

Contribution

It generalizes existing results for nonlinear schemes by applying a linearizing procedure and demonstrates the scheme's effectiveness and independence from local approximation order.

Findings

Existence of solutions under small time steps

Reduction in iteration count for the solution scheme

Numerical experiments confirming theoretical results

Abstract

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.