## Abstract

A new discontinuous Galerkin formulation is introduced for the elliptic reaction-diffusion problem that incorporates local second order distributional derivatives. The corresponding bilinear form satisfies both coercivity and continuity properties on the broken Hilbert space of H^{2} functions. For piecewise polynomial approximations of degree p {greater than or slanted equal to} 2, optimal uniform h and p convergence rates are obtained in the broken H^{1} and H^{2} norms. Convergence in L^{2} is optimal for p {greater than or slanted equal to} 3, if the computational mesh is strictly rectangular. If the mesh consists of skewed elements, then optimal convergence is only obtained if the corner angles satisfy a given regularity condition. For p = 2, only suboptimal h convergence rates in L^{2} are obtained and for linear polynomial approximations the method does not converge.

Original language | English (US) |
---|---|

Pages (from-to) | 3461-3482 |

Number of pages | 22 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 195 |

Issue number | 25-28 |

DOIs | |

State | Published - May 1 2006 |

## Keywords

- A priori error estimates
- Discontinuous Galerkin methods

## ASJC Scopus subject areas

- Computer Science Applications
- Computational Mechanics