Preconditioning by an extended matrix technique for convection-diffusion-reaction equations

Aurelian Nicola; Constantin Popa

doi:10.33993/jnaat372-890

Authors

Aurelian Nicola Ovidius University, Constanţa, Romania
Constantin Popa Ovidius University, Constanţa, Romania

DOI:

https://doi.org/10.33993/jnaat372-890

Keywords:

multilevel discretization, spectrally equivalent matrices, mesh independent preconditioning, positive semidefinite systems, CGLS algorithm

Abstract views: 196

Abstract

In this paper we consider a preconditioning technique for the ill-conditioned systems arising from discretisations of nonsymmetric elliptic boundary value problems. The rectangular preconditioning matrix is constructed via the transfer operators between successive discretization levels of the initial problem. In this way we get an extended, square, singular, consistent, but mesh independent well-conditioned linear system. Numerical experiments are presented for a 2D convection-diffusion-reaction problem.

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References

Björck, A., Numerical Methods for Least Squares Problems, SIAM Philadelphia, 1996, https://doi.org/10.1137/1.9781611971484 DOI: https://doi.org/10.1137/1.9781611971484

Briggs, L. W., A Multigrid Tutorial, SIAM Philadelphia, 1987.

Elman, H. C. and Schultz M. H., Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations, SIAM J. Numer. Anal., 23, no. 1, pp. 44-57, 1986, https://doi.org/10.1137/0723004 DOI: https://doi.org/10.1137/0723004

Golub, G. H. and van Loan, C. F., Matrix Computations, The John's Hopkins Univ. Press, Baltimore, 1983.

Griebel, M., Multilevel algorithms considered as iterative methods on semidefinite systems. SIAM J. Sci. Comput., 15, no. 3, pp. 547-565, 1994, https://doi.org/10.1137/0915036 DOI: https://doi.org/10.1137/0915036

Griebel, M., Zenger, C. and Zimmer, S., Multilevel Gauss-Seidel-algorithms for full and sparse grid problems. Computing, 50, pp. 127-148, 1993, https://doi.org/10.1007/bf02238610 DOI: https://doi.org/10.1007/BF02238610

Hackbusch, W., Elliptic Differential Equations. Theory and Numerical Treatment, Springer-Verlag, Berlin, 1987.

Hackbusch, W., Iterative Solution of Large Sparse Systems of Equations, Springer-Verlag, Berlin, 1994, https://doi.org/10.1007/978-1-4612-4288-8 DOI: https://doi.org/10.1007/978-1-4612-4288-8

Marchouk, G. and Agochkov, V., Introduction aux Méthodes des Éléments Finis, Éditions MIR, Moscou, 1985.

Oden, J.T. and Reddy, J.N., An Introduction to the Mathematical Theory of Finite Elements, John Wiley and Sons, Inc., 1976.

Oswald, P., Multilevel Finite Element Approximations, Teubner Skripten zur Numerik, Stuttgart, 1994. DOI: https://doi.org/10.1007/978-3-322-91215-2

Popa, C., Preconditioning conjugate gradient method for nonsymmetric systems, Intern. J. Computer Math., 58, pp. 117-133, 1995, https://doi.org/10.1080/00207169508804438 DOI: https://doi.org/10.1080/00207169508804438