A relatively new preconditioning technique called support graph preconditioning has
many merits over the traditional incomplete factorization based methods. A major
limitation of this technique is that it is applicable to symmetric diagonally dominant
matrices only. This work presents a technique that can be used to transform
the symmetric positive definite matrices arising from elliptic finite element problems
into symmetric diagonally dominant M-matrices. The basic idea is to approximate
the element gradient matrix by taking the gradients along chosen edges, whose unit
vectors form a new coordinate system. For Lagrangian elements, the rows of the
element gradient matrix in this new coordinate system are scaled edge vectors, thus
a diagonally dominant symmetric semidefinite M-matrix can be generated to approximate
the element stiffness matrix. Depending on the element type, one or more
such coordinate systems are required to obtain a global nonsingular M-matrix. Since
such approximation takes place at the element level, the degradation in the quality
of the preconditioner is only a small constant factor independent of the size of the
problem. This technique of element coordinate transformations applies to a variety of
first order Lagrangian elements. Combination of this technique and other techniques
enables us to construct an M-matrix preconditioner for a wide range of second order
elliptic problems even with higher order elements. Another contribution of this work is the proposal of a new variant of Vaidya’s
support graph preconditioning technique called modified domain partitioned support
graph preconditioners. Numerical experiments are conducted for various second order
elliptic finite element problems, along with performance comparison to the incomplete
factorization based preconditioners. Results show that these support graph preconditioners
are superior when solving ill-conditioned problems. In addition, the domain
partition feature provides inherent parallelism, and initial experiments show a good
potential of parallelization and scalability of these preconditioners.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-3135 |
| Date | 15 May 2009 |
| Creators | Wang, Meiqiu |
| Contributors | Sarin, Vivek |
| 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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