A Superconvergent Local Discontinuous Galerkin Method for Nonlinear Fourth-Order Boundary-Value Problems

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Abstract

In this paper, we present a superconvergent local discontinuous Galerkin (LDG) method for nonlinear fourth-order boundary-value problems (BVPs) of the form u(4) + f(x,u) = 0. We prove optimal L2 error estimates for the solution and for the three auxiliary variables that approximate the first, second, and third-order derivatives. The order of convergence is proved to be p + 1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. 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. Finally, we prove that the LDG solutions are superconvergent with order p + 3/2 toward Gauss-Radau projections of the exact solutions. Our numerical results indicate that the numerical order of superconvergence rate is p + 2. 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)
Article number1950035
JournalInternational Journal of Computational Methods
Volume17
Issue number7
DOIs
StatePublished - Sep 1 2020

Keywords

  • Fourth-order boundary-value problems
  • a priori error estimates
  • local discontinuous Galerkin method
  • superconvergence

ASJC Scopus subject areas

  • Computer Science (miscellaneous)
  • Computational Mathematics

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