Optimal error estimates and superconvergence of an ultra weak discontinuous Galerkin method for fourth-order boundary-value problems

Mahboub Baccouch, Helmi Temimi, Mohamed Ben-Romdhane

Research output: Contribution to journalArticle

Abstract

In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.

Original languageEnglish (US)
Pages (from-to)91-115
Number of pages25
JournalApplied Numerical Mathematics
Volume137
DOIs
StatePublished - Mar 2019

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Fourth-order Boundary Value Problem
Optimal Error Estimates
Discontinuous Galerkin
Superconvergence
Discontinuous Galerkin Method
Galerkin methods
Boundary value problems
Derivative
Optimal Rate of Convergence
Piecewise Polynomials
Nonlinear Boundary Value Problems
Linear Order
Polynomials
Derivatives
Mesh
Valid
Norm
Boundary conditions
Numerical Examples
Polynomial

Keywords

  • A priori error estimates
  • Fourth-order boundary-value problems
  • Superconvergence
  • Ultra weak discontinuous Galerkin method
  • Upwind and downwind points

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

Optimal error estimates and superconvergence of an ultra weak discontinuous Galerkin method for fourth-order boundary-value problems. / Baccouch, Mahboub; Temimi, Helmi; Ben-Romdhane, Mohamed.

In: Applied Numerical Mathematics, Vol. 137, 03.2019, p. 91-115.

Research output: Contribution to journalArticle

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N2 - In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.

AB - In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.

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