A superconvergent ultra-weak local discontinuous Galerkin method for nonlinear fourth-order boundary-value problems

Mahboub Baccouch

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form − u(4) = f(x, u). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the L2-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order p + 1 in the L2-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most p and mesh size h are used. We then show that the UWLDG solutions are superconvergent with order p + 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method.

Original languageEnglish (US)
Pages (from-to)1983-2023
Number of pages41
JournalNumerical Algorithms
Volume92
Issue number4
DOIs
StatePublished - Apr 2023

Keywords

  • A priori error estimate
  • Nonlinear fourth-order boundary-value problems
  • Superconvergence
  • Ultra-weak local discontinuous Galerkin method

ASJC Scopus subject areas

  • Applied Mathematics

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