### Abstract

This paper reviews some known and proposes some new preconditioning methods for a number of discontinuous Galerkin (or DG) finite element approximations for elliptic problems of second order. Nested hierarchy of meshes is generally assumed. Our approach utilizes a general two-level scheme, where the finite element space for the DG method is decomposed into a subspace (viewed as an auxiliary or 'coarse' space), plus a correction which can be handled by a standard smoothing procedure. We consider three different auxiliary subspaces, namely, piecewise linear C^{0}-conforming functions, piecewise linear functions that are continuous at the centroids of the edges/faces (Crouzeix-Raviart finite elements) and piecewise constant functions over the finite elements. To support the theoretical results, we also present numerical experiments for 3-D model problem showing uniform convergence of the constructed methods.

Original language | English (US) |
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Pages (from-to) | 753-770 |

Number of pages | 18 |

Journal | Numerical Linear Algebra with Applications |

Volume | 13 |

Issue number | 9 |

DOIs | |

State | Published - Nov 1 2006 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Applied Mathematics

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## Cite this

*Numerical Linear Algebra with Applications*,

*13*(9), 753-770. https://doi.org/10.1002/nla.504