### Abstract

Abstract In this paper, we analyze the discontinuous Galerkin (DG) finite element method for the steady two-dimensional transport-reaction equation on Cartesian grids. We prove the L^{2} stability and optimal L^{2} error estimates for the DG scheme. We identify a special numerical flux for which the L^{2}-norm of the solution is of order p + 1, when tensor product polynomials of degree at most p are used. We further prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be p + 1/2. We also provide a very simple derivative recovery formula which is O(hp+^{1}) superconvergent approximation to the directional derivative. Moreover, we establish an O(h2^{p+1}) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element. Finally, we prove that the proposed a posteriori DG error estimates converge to the true errors in the L^{2}-norm at O(hp+^{1}) rate and that the global effectivity index converges to unity at O(h) rate. Our results are valid without the flow condition restrictions. We perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal.

Original language | English (US) |
---|---|

Article number | 21119 |

Pages (from-to) | 144-162 |

Number of pages | 19 |

Journal | Applied Mathematics and Computation |

Volume | 265 |

DOIs | |

State | Published - May 27 2015 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

**Superconvergence and a posteriori error estimates of the DG method for scalar hyperbolic problems on Cartesian grids.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Superconvergence and a posteriori error estimates of the DG method for scalar hyperbolic problems on Cartesian grids

AU - Baccouch, Mahboub

PY - 2015/5/27

Y1 - 2015/5/27

N2 - Abstract In this paper, we analyze the discontinuous Galerkin (DG) finite element method for the steady two-dimensional transport-reaction equation on Cartesian grids. We prove the L2 stability and optimal L2 error estimates for the DG scheme. We identify a special numerical flux for which the L2-norm of the solution is of order p + 1, when tensor product polynomials of degree at most p are used. We further prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be p + 1/2. We also provide a very simple derivative recovery formula which is O(hp+1) superconvergent approximation to the directional derivative. Moreover, we establish an O(h2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element. Finally, we prove that the proposed a posteriori DG error estimates converge to the true errors in the L2-norm at O(hp+1) rate and that the global effectivity index converges to unity at O(h) rate. Our results are valid without the flow condition restrictions. We perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal.

AB - Abstract In this paper, we analyze the discontinuous Galerkin (DG) finite element method for the steady two-dimensional transport-reaction equation on Cartesian grids. We prove the L2 stability and optimal L2 error estimates for the DG scheme. We identify a special numerical flux for which the L2-norm of the solution is of order p + 1, when tensor product polynomials of degree at most p are used. We further prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be p + 1/2. We also provide a very simple derivative recovery formula which is O(hp+1) superconvergent approximation to the directional derivative. Moreover, we establish an O(h2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element. Finally, we prove that the proposed a posteriori DG error estimates converge to the true errors in the L2-norm at O(hp+1) rate and that the global effectivity index converges to unity at O(h) rate. Our results are valid without the flow condition restrictions. We perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal.

KW - A posteriori error estimates

KW - Cartesian grids

KW - Derivative recovery technique

KW - Discontinuous Galerkin method

KW - Hyperbolic problems

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=84930226833&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930226833&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2015.04.126

DO - 10.1016/j.amc.2015.04.126

M3 - Article

AN - SCOPUS:84930226833

VL - 265

SP - 144

EP - 162

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

M1 - 21119

ER -