Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations

Authors

Mahboub Baccouch, University of Nebraska at OmahaFollow

Document Type

Article

Publication Date

2016

Publication Title

Applied Numerical Mathematics

Abstract

We develop and analyze a new residual-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear ordinary differential equations (ODEs). The a posteriori DG error estimator under investigation is computationally simple, efficient, and asymptotically exact. It is obtained by solving a local residual problem with no boundary condition on each element. We first prove that the DG solution exhibits an optimal O(h^p+1) convergence rate in the L²-norm when p-degree piece-wise polynomials with p ≥1 are used. We further prove that the DG solution is O(h^2p+1) superconvergent at the downwind points. We use these results to prove that the p-degree DG solution is O(h^p+2) super close to a particular projection of the exact solution. This superconvergence result allows us to show that the true error can be divided into a significant part and a less significant part. The significant part of the discretization error for the DG solution is proportional to the (p +1)-degree right Radau polynomial and the less significant part converges at O(h^p+2) rate in the L²-norm. Numerical experiments demonstrate that the theoretical rates are optimal. Based on the global superconvergent approximations, we construct asymptotically exact a posteriori error estimates and prove that they converge to the true errors in the L²-norm under mesh refinement. The order of convergence is proved to be p +2. Finally, we prove that the global effectivity index in the L²-norm converges to unity at O(h)rate. Several numerical examples are provided to illustrate the global superconvergence results and the convergence of the proposed estimator under mesh refinement. A local adaptive procedure that makes use of our local a posteriori error estimate is also presented.

Comments

Published under Creative Commons license CC BY-NC-ND 4.0.

©2016 IMACS. Published by Elsevier B.V.

Recommended Citation

Baccouch, Mahboub, "Analysis of a posteriori error estimates of the discontinuous Galerkin method for nonlinear ordinary differential equations" (2016). Mathematics Faculty Publications. 5.
https://digitalcommons.unomaha.edu/mathfacpub/5