### Abstract

In this paper, we provide the first a posteriori error analysis of the discontinuous Galerkin (DG) method for solving the two-dimensional linear hyperbolic conservation laws on Cartesian grids. The key ingredients in our error analysis are the recent optimal superconvergence results proved in Cao et al. (SIAM J Numer Anal 53:1651–1671, 2015). We first prove that the DG solution converges in the L^{2}-norm to a Radau interpolating polynomial under mesh refinement. The order of convergence is proved to be p+ 2 , when tensor product polynomials of degree at most p are used. Then we show that the actual error can be divided into a significant part and a less significant part. The significant part of the DG error is spanned by two (p+ 1) -degree right Radau polynomials in the x and y directions. The less significant part converges to zero at O(h^{p} ^{+} ^{2}). These results are used to construct simple, efficient and asymptotically exact a posteriori error estimates. Superconvergence towards the right Radau interpolating polynomial is used to prove that, for smooth solutions, our a posteriori DG error estimates converge at a fixed time to the true spatial errors in the L^{2}-norm at O(h^{p} ^{+} ^{2}) rate. Finally, we prove that the global effectivity indices in the L^{2}-norm converge to unity at O(h) rate. Our proofs are valid for arbitrary regular Cartesian meshes using tensor product polynomials of degree at most p and for both the periodic and Dirichlet boundary conditions. Several numerical experiments are performed to validate the theoretical results.

Original language | English (US) |
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Pages (from-to) | 945-974 |

Number of pages | 30 |

Journal | Journal of Scientific Computing |

Volume | 68 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 2016 |

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### Keywords

- A posteriori error estimates
- Cartesian grids
- Discontinuous Galerkin method
- Hyperbolic problems
- Superconvergence

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Numerical Analysis
- Engineering(all)
- Computational Theory and Mathematics
- Computational Mathematics
- Applied Mathematics