We consider algorithms for preconditioning of two discontinuous Galerkin (DG)
methods for second order elliptic problems, namely the symmetric interior penalty
(SIPG) method and the method of Baumann and Oden.
For the SIPG method we first consider two-level preconditioners using coarse
spaces of either continuous piecewise polynomial functions or piecewise constant (discontinuous)
functions. We show that both choices give rise to uniform, with respect
to the mesh size, preconditioners. We also consider multilevel preconditioners based
on the same two types of coarse spaces. In the case when continuous coarse spaces
are used, we prove that a variable V-cycle multigrid algorithm is a uniform preconditioner.
We present numerical experiments illustrating the behavior of the considered
preconditioners when applied to various test problems in three spatial dimensions.
The numerical results confirm our theoretical results and in the cases not covered by
the theory show the efficiency of the proposed algorithms.
Another approach for preconditioning the SIPG method that we consider is an
algebraic multigrid algorithm using coarsening based on element agglomeration which
is suitable for unstructured meshes. We also consider an improved version of the algorithm
using a smoothed aggregation technique. We present numerical experiments
using the proposed algorithms which show their efficiency as uniform preconditioners.
For the method of Baumann and Oden we construct a preconditioner based on
an orthogonal splitting of the discrete space into piecewise constant functions and functions with zero average over each element. We show that the preconditioner
is uniformly spectrally equivalent to an appropriate symmetrization of the discrete
equations when quadratic or higher order finite elements are used. In the case of linear
elements we give a characterization of the kernel of the discrete system and present
numerical evidence that the method has optimal convergence rates in both L2 and
H1 norms. We present numerical experiments which show that the convergence of
the proposed preconditioning technique is independent of the mesh size.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2531 |
| Date | 15 May 2009 |
| Creators | Dobrev, Veselin Asenov |
| Contributors | Lazarov, Raytcho |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
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