### 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) |
---|---|

Pages (from-to) | 945-974 |

Number of pages | 30 |

Journal | Journal of Scientific Computing |

Volume | 68 |

Issue number | 3 |

DOIs | |

State | 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

### Cite this

**A Posteriori Error Analysis of the Discontinuous Galerkin Method for Two-Dimensional Linear Hyperbolic Conservation Laws on Cartesian Grids.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A Posteriori Error Analysis of the Discontinuous Galerkin Method for Two-Dimensional Linear Hyperbolic Conservation Laws on Cartesian Grids

AU - Baccouch, Mahboub

PY - 2016/9/1

Y1 - 2016/9/1

N2 - 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 L2-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(hp + 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 L2-norm at O(hp + 2) rate. Finally, we prove that the global effectivity indices in the L2-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.

AB - 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 L2-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(hp + 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 L2-norm at O(hp + 2) rate. Finally, we prove that the global effectivity indices in the L2-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.

KW - A posteriori error estimates

KW - Cartesian grids

KW - Discontinuous Galerkin method

KW - Hyperbolic problems

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=84955609755&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84955609755&partnerID=8YFLogxK

U2 - 10.1007/s10915-016-0166-0

DO - 10.1007/s10915-016-0166-0

M3 - Article

AN - SCOPUS:84955609755

VL - 68

SP - 945

EP - 974

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 3

ER -