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A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

✍ Scribed by Slimane Adjerid; Karen D. Devine; Joseph E. Flaherty; Lilia Krivodonova

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 510 KB
Volume: 191
Category: Article
ISSN: 0045-7825
DOI: 10.1016/s0045-7825(01)00318-8

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✦ Synopsis

We analyze the spatial discretization errors associated with solutions of one-dimensional hyperbolic conservation laws by discontinuous Galerkin methods (DGMs) in space. We show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a Radau polynomial of degree p 1 on each element. We also prove that the local and global discretization errors are ODx 2p1 and ODx 2p1 at the downwind point of each element. This strong superconvergence enables us to show that local and global discretization errors converge as ODx p2 at the remaining roots of Radau polynomial of degree p 1 on each element. Convergence of local and global discretization errors to the Radau polynomial of degree p 1 also holds for smooth solutions as p 3 I. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth.

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