### Abstract

In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler-Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to (k+1) -degree Radau polynomials, when polynomials of total degree not exceeding k are used. These results allow us to prove that the k -degree LDG solution and its derivatives are O(h ^{k+3/2}) superconvergent at the roots of (k+1) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time tconverge to the true errors at O(h^{k+5/4} rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the L^{2} -norm converge to unity at O(h^{1/2}) rate. Our proofs are valid for arbitrary regular meshes and for P^{k} polynomials with k ≥ 1, and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.

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

Pages (from-to) | 1-34 |

Number of pages | 34 |

Journal | Journal of Scientific Computing |

Volume | 60 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2014 |

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### Keywords

- A posteriori error estimates
- Adaptive mesh method
- Fourth-order Euler-Bernoulli equation
- Local discontinuous Galerkin method
- Radau points
- Superconvergence

### ASJC Scopus subject areas

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

### Cite this

**The local discontinuous galerkin method for the fourth-order euler-bernoulli partial differential equation in one space dimension. Part II : A posteriori error estimation.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The local discontinuous galerkin method for the fourth-order euler-bernoulli partial differential equation in one space dimension. Part II

T2 - A posteriori error estimation

AU - Baccouch, Mahboub

PY - 2014/7

Y1 - 2014/7

N2 - In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler-Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to (k+1) -degree Radau polynomials, when polynomials of total degree not exceeding k are used. These results allow us to prove that the k -degree LDG solution and its derivatives are O(h k+3/2) superconvergent at the roots of (k+1) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time tconverge to the true errors at O(hk+5/4 rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the L2 -norm converge to unity at O(h1/2) rate. Our proofs are valid for arbitrary regular meshes and for Pk polynomials with k ≥ 1, and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.

AB - In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler-Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to (k+1) -degree Radau polynomials, when polynomials of total degree not exceeding k are used. These results allow us to prove that the k -degree LDG solution and its derivatives are O(h k+3/2) superconvergent at the roots of (k+1) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time tconverge to the true errors at O(hk+5/4 rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the L2 -norm converge to unity at O(h1/2) rate. Our proofs are valid for arbitrary regular meshes and for Pk polynomials with k ≥ 1, and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.

KW - A posteriori error estimates

KW - Adaptive mesh method

KW - Fourth-order Euler-Bernoulli equation

KW - Local discontinuous Galerkin method

KW - Radau points

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=84902296036&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902296036&partnerID=8YFLogxK

U2 - 10.1007/s10915-013-9783-z

DO - 10.1007/s10915-013-9783-z

M3 - Article

AN - SCOPUS:84902296036

VL - 60

SP - 1

EP - 34

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 1

ER -