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 “high” and “low” 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.
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References
Aarnes, J.E.: Modelling of multiscale structures in flow simulations for petroleum reservoirs. In: Geometric Modelling, Numerical Simulation, and Optimization Applied Mathematics at SINTEF, pp. 307–360. Springer, New York (2007)
Aarnes, J.E., Hou, T.Y.: Multiscale domain decomposition methods for elliptic problems with high aspect ratios. Acta Math. Appl. Sin. 18(1), 63–76 (2002)
Aksoylu, B., Klie, H.: A family of physics-based preconditioners for solving elliptic equations on highly heterogeneous media. Appl. Numer. Mathe., 2008 (in press)
Chan, T.F., Mathew, T.: Domain decomposition methods. In Acta Numerica. Cambridge University Press, London (2004)
Christie, M.A., Blunt, M.J.: Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. SPE Reserv. Eng. 12, 308–317 (2001)
Cliffe, K.A., Graham, I.G., Scheichl, R., Stals, L.: Parallel computation of flow in heterogeneous media modelled by mixed finite elements. J. Comp. Phys. 164, 258–282 (2000)
Gerritsen, M.G., Durlofsky, L.J.: Modeling fluid flow in oil reservoirs. Annu. Rev. Fluid. Mech 37, 211–238 (2005)
Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore (1989)
Graham, I.G., Hagger, M.J.: Additive Schwarz, CG and discontinuous coefficients. In: Espedal, M., Bjørstad, P., Keyes, D.E. (eds.) Proc. 9th Intern. Confer. on Domain Decomposition Methods. Bergen, Norway (1998)
Graham, I.G., Hagger, M.J.: Unstructured additive Schwarz- conjugate gradient method for elliptic problems with highly discontinuous coefficients. SIAM J. Sci. Comp. 20(6), 2041–2066 (1999)
Graham, I.G., Lechner, P., Scheichl, R.: Domain decomposition for multiscale PDEs. Numer. Math. 106, 589–626 (2007)
Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations. Springer, New York (1993)
Khoromskij, B.N., Wittum, G.: Numerical Solution of Elliptic Differential Equations by Reduction to the Interface. In: Lecture Notes in Computer Science and Engineering, vol. 36. Springer, New York (2004)
Klie, H.: Krylov-Secant methods for solving Large Scale Systems of Coupled Nonlinear Parabolic Equations. Ph.D. thesis, Dept. of Computational and Applied Mathematics, Rice University, Houston, TX (1996)
Ming, P., Ye, X.: Numerical methods for multiscale elliptic problems. J. Comput. Phys. 214, 421–445 (2006)
Nabben, R., Vuik, C.: A comparison of deflation and the balancing preconditioner. SIAM J. Sci. Comp. 27, 1742–1759 (2006)
Nœtinger, B., Artus, V., Zargar, G.: The future of stochastic and upscaling methods in hydrogeology. Hydrogeol. J. 13, 184–201 (2005)
Brezina, M., Vanek, P., Mandel, J.: Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math. 88, 559–579 (2001)
Ruge, J.W., Stüben, K.: Algebraic multigrid. In: Multigrid Methods, Frontiers Appl. Math., vol. 3, pp. 73–130. SIAM, Philadelphia (1987)
Scheibe, T., Yabusaki, S.: Scaling of flow and transport behavior in heterogeneous groundwater systems. Adv. Water Resour. 22, 223–238 (1998)
Scheichl, R.: A rigorously justified algebraic preconditioner for high-contrast diffusion problems. in preparation, 2008
Scheichl, R., Vainikko, E.: Additive Schwarz and aggregation-based coarsening for elliptic problems with highly variable coefficients. Computing 80(4), 319–343 (2007)
Vuik, C., Segal, A., Meijerink, J.A.: An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts of coefficients. J. Comp. Phys. 152, 385–403 (1999)
Guadagnini, A., Sanchez-Vila, X., Carrera, J.: Representative hydraulic conductivities in saturated groundwater flow. Rev. Geophys. 44, 1–46 (2006)
Durlofsky, L.J., Wen, X.H., Edwards, M.G.: Upscaling of channel systems in two dimensions using flow-based grids. Transp. Porous Media 51, 343–366 (2003)
Xu, J., Zhu, Y.: Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Technical Report Technical Report No. AM311, Dept of Mathematics, Penn State, Pennsilvania, USA (2007)
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Communicated by G. Wittum.
Dedicated to Wolfgang Hackbusch on the occasion of his 60th birthday.
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Aksoylu, B., Graham, I.G., Klie, H. et al. Towards a rigorously justified algebraic preconditioner for high-contrast diffusion problems. Comput. Visual Sci. 11, 319–331 (2008). https://doi.org/10.1007/s00791-008-0105-1
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DOI: https://doi.org/10.1007/s00791-008-0105-1
Keywords
- Diffusion problem
- High-contrast coefficients
- Finite element approximation
- Algebraic preconditioner
- Schur complement
- Multigrid