A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems

Abstract

. In this paper, we present and analyze a posteriori error estimates in the L 2 -norm of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order boundary-value problems for ordinary di(cid:11)erential equations of the form u ′′ = f ( x;u ). We (cid:12)rst use the superconvergence results proved in the (cid:12)rst part of this paper ( J. Appl. Math. Comput. 69, 1507-1539, 2023) to prove that the UWDG solution converges, in the L 2 -norm, towards a special p -degree interpolating polynomial, when piecewise polynomials of degree at most p (cid:21) 2 are used. The order of convergence is proved to be p + 2. We then show that the UWDG error on each element can be divided into two parts. The dominant part is proportional to a special ( p +1)-degree Baccouch polynomial, which can be written as a linear combination of Legendre polynomials of degrees p (cid:0) 1, p , and p + 1. The second part converges to zero with order p + 2 in the L 2 - norm. These results allow us to construct a posteriori UWDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the L 2 -norm under mesh re(cid:12)nement. The order of convergence is proved to be p + 2. Finally, we prove that the global e(cid:11)ectivity index converges to unity at O ( h ) rate. Numerical results are presented exhibiting the reliability and the e(cid:14)ciency of the proposed error estimator.