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- W4313487649 abstract "Abstract Note: Please see pdf for full abstract with equations. 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′′ = ƒ(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 2,0 p−1 (ξ), ξ ∈ [−1, 1] , where P 2,0 p−1 (ξ) is the (p − 1 )-degree Jacobi polynomial, when piecewise polynomials of degree at most 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, 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. Several numerical experiments are provided to validate the theoretical results. AMS subject classifications (2020)}: 65L05, 65L20, 65L50, 65L60, 65L70." @default.
- W4313487649 created "2023-01-06" @default.
- W4313487649 creator A5068341694 @default.
- W4313487649 date "2023-01-04" @default.
- W4313487649 modified "2023-09-26" @default.
- W4313487649 title "Residual-based a posteriori error analysis of an ultra-weak discontinuous Galerkin method for nonlinear second-order initial-value problems" @default.
- W4313487649 doi "https://doi.org/10.21203/rs.3.rs-2420548/v1" @default.
- W4313487649 hasPublicationYear "2023" @default.
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