Least-squares finite element method for ordinary differential equations

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