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

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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(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.

Original languageEnglish (US)
Pages (from-to)1-34
Number of pages34
JournalJournal of Scientific Computing
Volume60
Issue number1
DOIs
StatePublished - Jul 2014

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Local Discontinuous Galerkin Method
A Posteriori Error Estimation
Galerkin methods
Bernoulli
Error analysis
Partial differential equations
Fourth Order
Euler
Partial differential equation
Discontinuous Galerkin
A Posteriori Error Estimates
Derivative
Polynomial
Polynomials
Derivatives
Error Estimates
Adaptive Procedure
Optimal Error Estimates
Superconvergence
Discretization Error

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

@article{0fdfad7e001a42f8921aea13c38774db,
title = "The local discontinuous galerkin method for the fourth-order euler-bernoulli partial differential equation in one space dimension. Part II: A posteriori error estimation",
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(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.",
keywords = "A posteriori error estimates, Adaptive mesh method, Fourth-order Euler-Bernoulli equation, Local discontinuous Galerkin method, Radau points, Superconvergence",
author = "Mahboub Baccouch",
year = "2014",
month = "7",
doi = "10.1007/s10915-013-9783-z",
language = "English (US)",
volume = "60",
pages = "1--34",
journal = "Journal of Scientific Computing",
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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.

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KW - Radau points

KW - Superconvergence

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