Superconvergence of the discontinuous Galerkin method for nonlinear second-order initial-value problems for ordinary differential equations

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Abstract

In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solutions are O(h2p+1) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.

Original languageEnglish (US)
Pages (from-to)160-179
Number of pages20
JournalApplied Numerical Mathematics
Volume115
DOIs
StatePublished - May 1 2017

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Discontinuous Galerkin
Superconvergence
Initial value problems
Discontinuous Galerkin Method
Piecewise Polynomials
Order of Convergence
Galerkin methods
Ordinary differential equations
Initial Value Problem
Ordinary differential equation
A Priori Error Estimates
Optimal Error Estimates
Auxiliary Variables
Arbitrary
Polynomials
Computational Results
Exact Solution
Mesh
Projection
Valid

Keywords

  • A priori error estimates
  • Discontinuous Galerkin method
  • Nonlinear second-order ordinary differential equations equation
  • Superconvergence

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

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title = "Superconvergence of the discontinuous Galerkin method for nonlinear second-order initial-value problems for ordinary differential equations",
abstract = "In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solutions are O(h2p+1) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.",
keywords = "A priori error estimates, Discontinuous Galerkin method, Nonlinear second-order ordinary differential equations equation, Superconvergence",
author = "Mahboub Baccouch",
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T1 - Superconvergence of the discontinuous Galerkin method for nonlinear second-order initial-value problems for ordinary differential equations

AU - Baccouch, Mahboub

PY - 2017/5/1

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N2 - In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solutions are O(h2p+1) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.

AB - In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solutions are O(h2p+1) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.

KW - A priori error estimates

KW - Discontinuous Galerkin method

KW - Nonlinear second-order ordinary differential equations equation

KW - Superconvergence

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