### Abstract

In this paper we present new superconvergence results for the local discontinuous Galerkin (LDG) method applied to the second-order scalar wave equation in one space dimension. Numerical experiments show O(hp+1)L2 convergence rate for the LDG solution and O(h ^{p+2}) superconvergent solutions at Radau points. More precisely, a local error analysis reveals that, at a fixed time t, the leading terms of the discretization errors for the solution and its derivative using p-degree polynomial approximations are proportional to the (p+1)-degree right Radau and (p+1)-degree left Radau polynomials, respectively. Thus, the p-degree LDG solution is O(h ^{p+2}) superconvergent at the roots of the (p+1)-degree right Radau polynomial and the derivative of the LDG solution is O(h ^{p+2}) superconvergent at the roots of the (p+1)-degree left Radau polynomial. These results are used to construct simple, efficient, and asymptotically correct a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement.

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

Pages (from-to) | 129-143 |

Number of pages | 15 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 209-212 |

DOIs | |

State | Published - Feb 1 2012 |

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### Keywords

- A posteriori error estimation
- Local discontinuous Galerkin method
- Second-order wave equation
- Superconvergence

### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials
- Mechanical Engineering
- Physics and Astronomy(all)
- Computer Science Applications

### Cite this

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

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A local discontinuous Galerkin method for the second-order wave equation

AU - Baccouch, Mahboub

PY - 2012/2/1

Y1 - 2012/2/1

N2 - In this paper we present new superconvergence results for the local discontinuous Galerkin (LDG) method applied to the second-order scalar wave equation in one space dimension. Numerical experiments show O(hp+1)L2 convergence rate for the LDG solution and O(h p+2) superconvergent solutions at Radau points. More precisely, a local error analysis reveals that, at a fixed time t, the leading terms of the discretization errors for the solution and its derivative using p-degree polynomial approximations are proportional to the (p+1)-degree right Radau and (p+1)-degree left Radau polynomials, respectively. Thus, the p-degree LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree right Radau polynomial and the derivative of the LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree left Radau polynomial. These results are used to construct simple, efficient, and asymptotically correct a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement.

AB - In this paper we present new superconvergence results for the local discontinuous Galerkin (LDG) method applied to the second-order scalar wave equation in one space dimension. Numerical experiments show O(hp+1)L2 convergence rate for the LDG solution and O(h p+2) superconvergent solutions at Radau points. More precisely, a local error analysis reveals that, at a fixed time t, the leading terms of the discretization errors for the solution and its derivative using p-degree polynomial approximations are proportional to the (p+1)-degree right Radau and (p+1)-degree left Radau polynomials, respectively. Thus, the p-degree LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree right Radau polynomial and the derivative of the LDG solution is O(h p+2) superconvergent at the roots of the (p+1)-degree left Radau polynomial. These results are used to construct simple, efficient, and asymptotically correct a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori errors estimates under mesh refinement.

KW - A posteriori error estimation

KW - Local discontinuous Galerkin method

KW - Second-order wave equation

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=83455262231&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=83455262231&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2011.10.012

DO - 10.1016/j.cma.2011.10.012

M3 - Article

AN - SCOPUS:83455262231

VL - 209-212

SP - 129

EP - 143

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

ER -