Residual-based a posteriori error analysis of an ultra-weak discontinuous Galerkin method for nonlinear second-order initial-value problems

In this paper, we propose and analyze implicit residual-based a posteriori error estimates for the ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form \(u''=f(x,u)\) $u^{\prime \prime }=f(x,u)$ . We prove that the UWDG error on each element can be split into two parts. The significant part is proportional to the \((p+1)\) $(p+1)$ -degree polynomial \((1-\xi )^2P_{p-1}^{2,0}(\xi ),\ \xi \in [-1,1]\) ${(1-\xi )}^2{P}_{p-1}^{2,0}(\xi ),\xi \in [-1,1]$ , where \(P_{p-1}^{2,0}(\xi )\) $P_{p-1}^{2,0}(\xi )$ is the \((p-1)\) $(p-1)$ -degree Jacobi polynomial, when piecewise polynomials of degree \(p\ge 2\) $p\ge 2$ are used. The second part of the error converges with order \(p+2\) $p+2$ in the \(L^2\) $L^2$ -norm. These results allow us to construct a posteriori UWDG error estimates. The proposed residual-based a posteriori error estimators of this paper are reliable and efficient and are obtained by solving a local problem with no side conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the \(L^2\) $L^2$ -norm under mesh refinement. The order of convergence is proved to be \(p+2\) $p+2$ . Finally, we prove that the global effectivity index converges to unity at \(\mathcal {O}(h)\) $\mathcal{O}(h)$ rate. As an application, we introduce a local adaptive mesh refinement (AMR) procedure that leverages both our local and global a posteriori error estimates. Our proofs hold for general regular meshes and for \(P^p\) $P^p$ polynomials with \(p\ge 1\) $p\ge 1$ . Several numerical experiments are provided to validate the theoretical results.