Least-Squares Finite Element Method for Ordinary Differential Equations

by   Matthias Chung, et al.

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.



There are no comments yet.


page 1

page 2

page 3

page 4


On the development of symmetry-preserving finite element schemes for ordinary differential equations

In this paper we introduce a procedure, based on the method of equivaria...

Numerical Integration as an Initial Value Problem

Numerical integration (NI) packages commonly used in scientific research...

A discontinuous least squares finite element method for time-harmonic Maxwell equations

We propose and analyze a discontinuous least squares finite element meth...

Efficient Magnus-type integrators for solar energy conversion in Hubbard models

Strongly interacting electrons in solids are generically described by Hu...

A simple third order compact finite element method for 1D BVP

A simple third order compact finite element method is proposed for one-d...

Convergence analysis of collocation methods for computing periodic solutions of retarded functional differential equations

We analyze the convergence of piecewise collocation methods for computin...

GSEIM: A General-purpose Simulator with Explicit and Implicit Methods

A new simulation package, GSEIM, for solving a set of ordinary different...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.