### Abstract

In this paper, we propose and analyze a new superconvergent local discontinuous Galerkin (LDG) method equipped with an element residual error estimator for the spatial discretization of the second-order wave equation on Cartesian grids. We prove the L^{2} stability, the energy conserving property, and optimal L^{2} error estimates for the semi-discrete formulation. In particular, we identify special numerical fluxes for which the L^{2}-norm of the solution and its gradient are of order p+1, when tensor product polynomials of degree at most p are used. We further perform a local error analysis and show that the leading term of the LDG error is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, the LDG solution is O(h^{p+2}) superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree right Radau polynomial. Furthermore, numerical computations show that the first component of the solution's gradient is O(h^{p+2}) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h^{p+2}) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Computational results indicate that global superconvergence holds for LDG solutions. We use the superconvergence results to construct a posteriori LDG error estimates. These error estimates are computationally simple and are obtained by solving local steady problems with no boundary condition on each element. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement.

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

Pages (from-to) | 1250-1278 |

Number of pages | 29 |

Journal | Computers and Mathematics with Applications |

Volume | 68 |

Issue number | 10 |

DOIs | |

State | Published - Nov 1 2014 |

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### Keywords

- A posteriori error estimates
- Cartesian grids
- Local discontinuous Galerkin methods
- Second-order wave equation
- Superconvergence

### ASJC Scopus subject areas

- Modeling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

**A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

AU - Baccouch, Mahboub

PY - 2014/11/1

Y1 - 2014/11/1

N2 - In this paper, we propose and analyze a new superconvergent local discontinuous Galerkin (LDG) method equipped with an element residual error estimator for the spatial discretization of the second-order wave equation on Cartesian grids. We prove the L2 stability, the energy conserving property, and optimal L2 error estimates for the semi-discrete formulation. In particular, we identify special numerical fluxes for which the L2-norm of the solution and its gradient are of order p+1, when tensor product polynomials of degree at most p are used. We further perform a local error analysis and show that the leading term of the LDG error is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, the LDG solution is O(hp+2) superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree right Radau polynomial. Furthermore, numerical computations show that the first component of the solution's gradient is O(hp+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(hp+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Computational results indicate that global superconvergence holds for LDG solutions. We use the superconvergence results to construct a posteriori LDG error estimates. These error estimates are computationally simple and are obtained by solving local steady problems with no boundary condition on each element. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement.

AB - In this paper, we propose and analyze a new superconvergent local discontinuous Galerkin (LDG) method equipped with an element residual error estimator for the spatial discretization of the second-order wave equation on Cartesian grids. We prove the L2 stability, the energy conserving property, and optimal L2 error estimates for the semi-discrete formulation. In particular, we identify special numerical fluxes for which the L2-norm of the solution and its gradient are of order p+1, when tensor product polynomials of degree at most p are used. We further perform a local error analysis and show that the leading term of the LDG error is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, the LDG solution is O(hp+2) superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree right Radau polynomial. Furthermore, numerical computations show that the first component of the solution's gradient is O(hp+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(hp+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Computational results indicate that global superconvergence holds for LDG solutions. We use the superconvergence results to construct a posteriori LDG error estimates. These error estimates are computationally simple and are obtained by solving local steady problems with no boundary condition on each element. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement.

KW - A posteriori error estimates

KW - Cartesian grids

KW - Local discontinuous Galerkin methods

KW - Second-order wave equation

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=84908432090&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908432090&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2014.08.023

DO - 10.1016/j.camwa.2014.08.023

M3 - Article

VL - 68

SP - 1250

EP - 1278

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 10

ER -