Support graph preconditioning for elliptic finite element problems

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Title: Support graph preconditioning for elliptic finite element problems
Author: Wang, Meiqiu
Abstract: 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 .
URI: http : / /hdl .handle .net /1969 .1 /ETD -TAMU -3135
Date: 2009-05-15


Support graph preconditioning for elliptic finite element problems. Available electronically from http : / /hdl .handle .net /1969 .1 /ETD -TAMU -3135 .

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