Least-Squares Finite Element Method for Ordinary Differential Equations

TL;DR

This paper establishes optimal error estimates for the least-squares finite element method applied to nonlinear ODEs, demonstrating its advantages over finite difference methods through numerical examples and discussing adaptive time-stepping modifications.

Contribution

It provides the first optimal error analysis for lsfem on nonlinear ODEs with piecewise linear elements and explores extensions to higher-order basis functions.

Findings

Optimal error estimates are proven for lsfem with linear elements.

Numerical experiments confirm theoretical results and show advantages over finite difference methods.

Extensions to higher-order basis elements and adaptive time-stepping are discussed.

Abstract

We consider the least-squares finite element method (lsfem) for systems of nonlinear ordinary differential equations and establish an optimal error estimate for this method when piecewise linear elements are used. The main assumptions are that the vector field is sufficiently smooth and that the local Lipschitz constant, as well as the operator norm of the Jacobian matrix associated with the nonlinearity, are sufficiently small when restricted to a suitable neighborhood of the true solution for the considered initial value problem. This theoretic optimality is further illustrated numerically, along with evidence of possible extension to higher-order basis elements. Examples are also presented to show the advantages of lsfem compared with finite difference methods in various scenarios. Suitable modifications for adaptive time-stepping are discussed as well.