Analysis of optimal superconvergence of a local discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

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Abstract

In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u=f(x,u,u), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. 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 derivatives of the LDG solutions are superconvergent with order p+1 toward the derivatives of Gauss-Radau projections of the exact solutions. Moreover, we prove that the LDG solutions are superconvergent with order p+2 toward Gauss-Radau projections of the exact solutions. Finally, we prove, for any polynomial degree p, the (2p+1)th superconvergence rate of the LDG approximations at the upwind or downwind points and for the domain average under quasi-uniform meshes. Our numerical experiments demonstrate optimal rates of convergence and superconvergence. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p≥1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

Original languageEnglish (US)
Pages (from-to)361-383
Number of pages23
JournalApplied Numerical Mathematics
Volume145
DOIs
Publication statusPublished - Nov 2019

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Keywords

  • A priori error estimates
  • Local discontinuous Galerkin method
  • Nonlinear second-order boundary-value problems
  • Superconvergence

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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