Abstract
We discuss the preconditioning of systems coupling elliptic operators in \(\Omega \subset {\mathbb{R}}^{d}\), d=2,3, with elliptic operators defined on hypersurfaces. These systems arise naturally when physical phenomena are affected by geometric boundary forces, such as the evolution of liquid drops subject to surface tension. The resulting operators are sums of interior and boundary terms weighted by parameters. We investigate the behavior of multigrid algorithms suited to this context and demonstrate numerical results which suggest uniform preconditioning bounds that are level and parameter independent.
Mathematics Subject Classification (2010): 65N30, 65N55
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Notes
- 1.
λ 0 is the smallest eigenvalue of the preconditioned system.
References
Bänsch, E.: Finite element discretization of the Navier-Stokes equations with a free capillary surface. Numer. Math. 88(2), 203–235 (2001)
Bonito, A., Pasciak, J.E.: Analysis of a multigrid algorithm for an elliptic problem with a perturbed boundary condition. Technical report (in preparation)
Bonito, A., Pasciak, J.E.: Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator. Math. Comp. 81(279), 1263–1288 (2012)
Bonito, A., Nochetto, R.H., Pauletti, M.S.: Dynamics of biomembranes: effect of the bulk fluid. Math. Model. Nat. Phenom. 6(5), 25–43 (2011)
Bramble, J.H., Pasciak, J.E.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comp. 50(181), 1–17 (1988)
Bramble, J.H., Pasciak, J.E.: The analysis of smoothers for multigrid algorithms. Math. Comp. 58(198), 467–488 (1992)
Bramble, J., Zhang, X.: The analysis of multigrid methods. In: Ciarlet, P.C., Lions, J.L. (eds.) Handbook of Numerical Analysis, Techniques of Scientific Computing (Part 3). Elsevier, Amsterdam (2000)
Bramble, J.H., Pasciak, J.E., Vassilev, A.T.: Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34(3), 1072–1092 (1997)
Bramble, J.H., Pasciak, J.E., Vassilevski, P.S.: Computational scales of Sobolev norms with application to preconditioning. Math. Comp. 69(230), 463–480 (2000)
Du, Q., Liu, Ch., Ryham, R., Wang, X.: Energetic variational approaches in modeling vesicle and fluid interactions. Phys. D 238(9–10), 923–930 (2009)
Dziuk, G.: An algorithm for evolutionary surfaces. Numer. Math. 58(6), 603–611 (1991)
Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition
Haase, G., Langer, U., Meyer, A., Nepomnyaschikh, S.V.: Hierarchical extension operators and local multigrid methods in domain decomposition preconditioners. East-West J. Numer. Math. 2(3), 173–193 (1994)
Hysing, S.: A new implicit surface tension implementation for interfacial flows. Int. J. Numer. Methods Fluids 51(6), 659–672 (2006)
Lee, Y.-Ju., Wu, J., Xu, J., Zikatanov, L.: Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci. 17(11), 1937–1963 (2007)
Lee, Y.-Ju., Wu, J., Xu, J., Zikatanov, L.: A sharp convergence estimate for the method of subspace corrections for singular systems of equations. Math. Comp. 77(262), 831–850 (2008)
Pauletti, M.S.: Parametric AFEM for geometric evolution equation and coupled fluid-membrane interaction. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.), University of Maryland, College Park (2008)
Rusten, T., Winther, R.: A preconditioned iterative method for saddlepoint problems. SIAM J. Matrix Anal. Appl. 13(3), 887–904 (1992). Iterative Methods in Numerical Linear Algebra, Copper Mountain, CO (1990)
Sohn, J.S., Tseng, Y.-H., Li, S., Voigt, A., Lowengrub, J.S.: Dynamics of multicomponent vesicles in a viscous fluid. J. Comput. Phys. 229(1), 119–144, (2010)
Walker, S.W., Bonito, A., Nochetto, R.H.: Mixed finite element method for electrowetting on dielectric with contact line pinning. Interfaces Free Bound. 12(1), 85–119 (2010)
Acknowledgements
This work was supported in part by award number KUS-C1-016-04 made by King Abdulla University of Science and Technology (KAUST). It was also supported in part by the National Science Foundation through Grant DMS-0914977 and DMS-1216551.
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Bonito, A., Pasciak, J.E. (2013). A Multigrid Algorithm for an Elliptic Problem with a Perturbed Boundary Condition. In: Iliev, O., Margenov, S., Minev, P., Vassilevski, P., Zikatanov, L. (eds) Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications. Springer Proceedings in Mathematics & Statistics, vol 45. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7172-1_4
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