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 ) . We prove that the UWDG error on each element can be split into two parts. The significant part is proportional to the ( p + 1 ) -degree polynomial ( 1 - ξ ) 2 P p - 1 2 , 0 ( ξ ) , ξ ∈ [ - 1 , 1 ] , where P p - 1 2 , 0 ( ξ ) is the ( p - 1 ) -degree Jacobi polynomial, when piecewise polynomials of degree p ≥ 2 are used. The second part of the error converges with order p + 2 in the 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 -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 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 polynomials with p ≥ 1 . Several numerical experiments are provided to validate the theoretical results.