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

Superconvergence error analysis

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler-Bernoulli beam equation in one space dimension. We prove the $$L 2$$ L 2 stability of the scheme and several optimal L 2 error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are O (h k + 3 / 2) super close to particular projections of the exact solutions for k th-degree polynomial spaces while computational results show higher O (h k + 2) convergence rate. Our proofs are valid for arbitrary regular meshes and for P k polynomials with ≥ 1k 1, and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the L 2 -norm under mesh refinement.

Original languageEnglish (US)
Pages (from-to)795-840
Number of pages46
JournalJournal of Scientific Computing
Volume59
Issue number3
DOIs
StatePublished - Jan 1 2014

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Local Discontinuous Galerkin Method
Superconvergence
Galerkin methods
Bernoulli
Error Analysis
Error analysis
Partial differential equations
Fourth Order
Euler
Partial differential equation
Discontinuous Galerkin
Derivative
Error Estimates
Exact Solution
Derivatives
Projection
Beam Equation
Euler-Bernoulli Beam
Optimal Rate of Convergence
Polynomial

Keywords

  • Fourth-order Euler-Bernoulli equation
  • Local discontinuous Galerkin method
  • Optimal error estimates
  • Projection
  • Stability
  • Superconvergence

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Software
  • Engineering(all)
  • Computational Theory and Mathematics

Cite this

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title = "The local discontinuous galerkin method for the fourth-order euler-bernoulli partial differential equation in one space dimension. Part I: Superconvergence error analysis",
abstract = "In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler-Bernoulli beam equation in one space dimension. We prove the $$L 2$$ L 2 stability of the scheme and several optimal L 2 error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are O (h k + 3 / 2) super close to particular projections of the exact solutions for k th-degree polynomial spaces while computational results show higher O (h k + 2) convergence rate. Our proofs are valid for arbitrary regular meshes and for P k polynomials with ≥ 1k 1, and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the L 2 -norm under mesh refinement.",
keywords = "Fourth-order Euler-Bernoulli equation, Local discontinuous Galerkin method, Optimal error estimates, Projection, Stability, Superconvergence",
author = "Mahboub Baccouch",
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N2 - In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler-Bernoulli beam equation in one space dimension. We prove the $$L 2$$ L 2 stability of the scheme and several optimal L 2 error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are O (h k + 3 / 2) super close to particular projections of the exact solutions for k th-degree polynomial spaces while computational results show higher O (h k + 2) convergence rate. Our proofs are valid for arbitrary regular meshes and for P k polynomials with ≥ 1k 1, and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the L 2 -norm under mesh refinement.

AB - In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler-Bernoulli beam equation in one space dimension. We prove the $$L 2$$ L 2 stability of the scheme and several optimal L 2 error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are O (h k + 3 / 2) super close to particular projections of the exact solutions for k th-degree polynomial spaces while computational results show higher O (h k + 2) convergence rate. Our proofs are valid for arbitrary regular meshes and for P k polynomials with ≥ 1k 1, and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the L 2 -norm under mesh refinement.

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