A superconvergent local discontinuous Galerkin method for elliptic problems

Slimane Adjerid, Mahboub Baccouch

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

In this manuscript we investigate the convergence properties of a minimal dissipation local discontinuous Galerkin(md-LDG) method for two-dimensional diffusion problems on Cartesian meshes. Numerical computations show O(h p+1) L 2 convergence rates for the solution and its gradient and O(h p+2) superconvergent solutions at Radau points on enriched p-degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a p-degree discontinuous finite element solution is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (p+1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution's gradient is O(h p+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h p+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Several numerical simulations are performed to validate the theory.

Original languageEnglish (US)
Pages (from-to)113-152
Number of pages40
JournalJournal of Scientific Computing
Volume52
Issue number1
DOIs
StatePublished - Jul 1 2012

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Local Discontinuous Galerkin Method
Galerkin methods
Elliptic Problems
Polynomials
Polynomial
Tensor Product
Tensors
Roots
Discontinuous Finite Elements
Gradient
y direction
x direction
Diffusion Problem
Finite Element Solution
Cartesian
Error Analysis
Convergence Properties
Numerical Computation
Error analysis
Convergence Rate

Keywords

  • Elliptic problems
  • Local discontinuous Galerkin method
  • Superconvergence

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

Cite this

A superconvergent local discontinuous Galerkin method for elliptic problems. / Adjerid, Slimane; Baccouch, Mahboub.

In: Journal of Scientific Computing, Vol. 52, No. 1, 01.07.2012, p. 113-152.

Research output: Contribution to journalArticle

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