Abstract
In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale \(H(\mathrm {div})\) problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this \(H(\mathrm {div})\) problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart–Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale \(H(\mathrm {div})\) system is bounded by \(C(1+\frac{H^2}{{\hat{\delta }}^2})(1+\log ^4(\frac{H}{h}))\), where \(\hat{\delta }\) denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.
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Arbogast, T.: Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal. 42, 576–598 (2004)
Arbogast, T., Boyd, K.J.: Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44, 1150–1171 (2006)
Arnold, D.N., Falk, R.S., Winther, R.: Preconditioning in \(H(\text{div})\) and applications. Math. Comp. 66, 957–984 (1997)
Araya, R., Harder, C., Paredes, D., Valentin, F.: Multiscale hybrid-mixed method. SIAM J. Numer. Anal. 51, 3505–3531 (2013)
Babuška, I., Osborn, J.E.: Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20, 510–536 (1983)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)
Christie, M., Blunt, M.J.: Tenth SPE comparative solution project: a comparison of upscaling techniques. SPE Res. Eval. Eng. 4, 308–317 (2001)
Chung, E.T., Efendiev, Y., Lee, C.S.: Mixed generalized multiscale finite element methods and applications. Multiscale Model. Simul. 13, 338–366 (2015)
Chen, Z., Hou, T.Y.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp. 72, 541–576 (2003)
Chen, Z.X., Huan, G.R., Ma, Y.L.: Computational Methods for Multiphase Flows in Porous Media, vol. 2, SIAM, Philadelphia (2006)
Dolean, V., Nataf, F., Scheichl, R., Spillane, N.: Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps. Comput. Methods Appl. Math. 12, 391–414 (2012)
Elman, H.C., Golub, G.H.: Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31, 1645–1661 (1994)
Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2005)
Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251, 116–135 (2013)
Galvis, J., Efendiev, Y.: Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul. 8, 1461–1483 (2010)
Galvis, J., Efendiev, Y.: Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model. Simul. 8, 1621–1644 (2010)
Graham, I.G., Lechner, P.O., Scheichl, R.: Domain decomposition for multiscale PDEs. Numer. Math. 106, 589–626 (2007)
Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)
Hellman, F., Målqvist, A.: Contrast independent localization of a multiscale problems. Multiscale Model. Simul. 15, 1325–1355 (2017)
Hiptmair, R.: Multigrid method for H(\(\text{ div }\)) in three dimensions. Electron. Trans. Numer. Anal. 6, 133–152 (1997)
Hiptmair, R., Xu, J.: Nodal auxiliary space preconditioning in \(H(\text{curl})\) and \(H(\text{div})\) spaces. SIAM J. Numer. Anal. 45, 2483–2509 (2007)
Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)
Hu, Q.Y., Shu, S., Zou, J.: A discrete weighted Helmholtz decomposition and its application. Numer. Math. 125, 153–189 (2013)
Jenny, P., Lee, S.H., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187, 47–67 (2003)
Målqvist, A., Peterseim, D.: Localization of elliptic multiscale problems. Math. Comp. 83, 2583–2603 (2014)
Mathew, T.P.: Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part \({\text{ I }}\): algorithms and numerical results. Numer. Math. 65, 445–468 (1993)
Mathew, T.P.: Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part \(\text{ II }\): convergence theory. Numer. Math. 65, 469–492 (1993)
Oh, D.S.: An overlapping schwarz algorithm for Raviart–Thomas vector fields with discontinuous coefficients. SIAM J. Numer. Anal. 51, 297–321 (2013)
Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)
Rusten, T., Winther, R.: A preconditioned iterative method for saddlepoint problems. SIAM J. Matrix Anal. Appl. 13, 887–904 (1992)
Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Numer. Math. 126, 741–770 (2014)
Spillane, N., Rixen, D.J.: Automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms. Int. J. Numer. Methods Eng. 95, 953–990 (2013)
Toselli, A., Widlund, O.B.: Domain Decomposition Methods—Algorithms and Theory, vol. 34. Springer, New York (2005)
Vassilevski, P.S., Lazarov, R.D.: Preconditioning mixed finite element saddle-point elliptic problems. Numer. Linear Algebra Appl. 3, 1–20 (1996)
Vassilevski, P.S., Wang, J.P.: Multilevel iterative methods for mixed finite element discretizations of elliptic problems. Numer. Math. 63, 503–520 (1992)
Wohlmuth, B.I., Toselli, A., Widlund, O.B.: An iterative substructuring method for Raviart–Thomas vector fields in three dimensions. SIAM J. Numer. Anal. 37, 1657–1676 (2000)
Xie, H., Xu, X.: Mass conservative domain decomposition preconditioners for multiscale finite volume method. Multiscale Model. Simul. 12, 1667–1690 (2014)
Acknowledgements
The authors would like to thank the editor and anonymous referees who made many helpful comments and suggestions which lead to an improved presentation of this paper. The work of the second author was supported by the National Natural Science Foundation of China (No. 11671302).
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Xie, H., Xu, X. Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems. Commun. Appl. Math. Comput. 1, 141–165 (2019). https://doi.org/10.1007/s42967-019-0005-z
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DOI: https://doi.org/10.1007/s42967-019-0005-z