Volume 5, Issue 3
Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

Y. Efendiev, J. Galvis, S. Ki Kang & R.D. Lazarov

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 359-383.

Published online: 2012-05

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  • Abstract

In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results.

  • Keywords

FE method, nonlinear permeability, highly heterogeneous media, high contrast media.

  • AMS Subject Headings

65M60, 35Q35, 35Q86, 65N30, 65N55

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-5-359, author = {}, title = {Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {3}, pages = {359--383}, abstract = {

In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2012.m1112}, url = {http://global-sci.org/intro/article_detail/nmtma/5942.html} }
TY - JOUR T1 - Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 359 EP - 383 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2012.m1112 UR - https://global-sci.org/intro/article_detail/nmtma/5942.html KW - FE method, nonlinear permeability, highly heterogeneous media, high contrast media. AB -

In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results.

Y. Efendiev, J. Galvis, S. Ki Kang & R.D. Lazarov. (2020). Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media. Numerical Mathematics: Theory, Methods and Applications. 5 (3). 359-383. doi:10.4208/nmtma.2012.m1112
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