TY - JOUR
T1 - A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems
AU - Baccouch , Mahboub
JO - International Journal of Numerical Analysis and Modeling
VL - 5
SP - 618
EP - 646
PY - 2023
DA - 2023/09
SN - 20
DO - http://doi.org/10.4208/ijnam2023-1027
UR - https://global-sci.org/intro/article_detail/ijnam/22005.html
KW - Second-order boundary-value problems, ultra-weak discontinuous Galerkin method, superconvergence, a posteriori error estimation, Baccouch polynomials.
AB - <p style="text-align: justify;">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 differential equations of the form $u′′=f(x, u).$ We first use the
superconvergence results proved in the first 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 $p ≥ 2$ 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 $p − 1,$ $p,$ 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 refinement.
The order of convergence is proved to be $p + 2.$ Finally, we prove that the global effectivity index
converges to unity at $\mathcal{O}(h)$ rate. Numerical results are presented exhibiting the reliability and the
efficiency of the proposed error estimator.</p>