Abstract
In this paper, we propose and analyze a robust recovery-based error estimator for the original discontinuous Galerkin method for nonlinear scalar conservation laws in one space dimension. The proposed a posteriori error estimator of the recovery-type is easy to implement, computationally simple, asymptotically exact, and is useful in adaptive computations. We use recent results (Meng et al. in SIAM J Numer Anal 50:2336–2356, 2012) to prove that, for smooth solutions, our a posteriori error estimates at a fixed time converge to the true spatial errors in the \(L^2\)-norm under mesh refinement. The order of convergence is proved to be \(p + 1\), when \(p\)-degree piecewise polynomials with \(p\ge 1\) are used. We further prove that the global effectivity index converges to unity at \(\mathcal {O}(h)\) rate. Our proofs are valid for arbitrary regular meshes using \(P^p\) polynomials with \(p\ge 1\), under the condition that \(|f'(u)|\) possesses a uniform positive lower bound, where \(f(u)\) is the nonlinear flux function. We provide several numerical examples to support our theoretical results, to show the effectiveness of our recovery-based a posteriori error estimates, and to demonstrate that our results hold true for nonlinear conservation laws with general flux functions. These experiments indicate that the restriction on \(f(u)\) is artificial.
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The authors would like to thank the referee for the valuable comments and suggestions which improve the quality of the paper. This research was supported by the University Committee on Research and Creative Activity (UCRCA Proposal 2015-01-F) at the University of Nebraska at Omaha.
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Baccouch, M. Recovery-Based Error Estimator for the Discontinuous Galerkin Method for Nonlinear Scalar Conservation Laws in One Space Dimension. J Sci Comput 66, 459–476 (2016). https://doi.org/10.1007/s10915-015-0030-7
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DOI: https://doi.org/10.1007/s10915-015-0030-7