### 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 language | English (US) |
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Pages (from-to) | 113-152 |

Number of pages | 40 |

Journal | Journal of Scientific Computing |

Volume | 52 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 1 2012 |

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### 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