التفاصيل البيبلوغرافية
| العنوان: |
Towards a rigorously justified algebraic preconditioner for high-contrast diffusion problems. |
| المؤلفون: |
Burak Aksoylu, Ivan Graham, Hector Klie, Robert Scheichl |
| المصدر: |
Computing & Visualization in Science; Sep2008, Vol. 11 Issue 4-6, p319-331, 13p |
| مصطلحات موضوعية: |
NUMERICAL analysis, LINEAR systems, ALGORITHMS, MATRICES (Mathematics) |
| مستخلص: |
Abstract  In this paper we present a new preconditioner suitable for solving linear systems arising from finite element approximations of elliptic PDEs with high-contrast coefficients. The construction of the preconditioner consists of two phases. The first phase is an algebraic one which partitions the degrees of freedom into âhighâ and âlowâ permeability regions which may be of arbitrary geometry. This partition yields a corresponding blocking of the stiffness matrix and hence a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. The structure of the required sub-block inverses in the high contrast case is revealed by a singular perturbation analysis (with the contrast playing the role of a large parameter). This shows that for high enough contrast each of the sub-block inverses can be approximated well by solving only systems with constant coefficients. The second phase of the algorithm involves the approximation of these constant coefficient systems using multigrid methods. The result is a general method of algebraic character which (under suitable hypotheses) can be proved to be robust with respect to both the contrast and the mesh size. While a similar performance is also achieved in practice by algebraic multigrid (AMG) methods, this performance is still without theoretical justification. Since the first phase of our method is comparable to the process of identifying weak and strong connections in conventional AMG algorithms, our theory provides to some extent a theoretical justification for these successful algebraic procedures. We demonstrate the advantageous properties of our preconditioner using experiments on model problems. Our numerical experiments show that for sufficiently high contrast the performance of our new preconditioner is almost identical to that of the Ruge and Stüben AMG preconditioner, both in terms of iteration count and CPU-time. [ABSTRACT FROM AUTHOR] |
|
Copyright of Computing & Visualization in Science is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) |
| قاعدة البيانات: |
Complementary Index |
Find this article in full text from Springer Verlag. 
Full Text Finder