Skip to main content
SpringerLink
Log in

Substructuring Preconditioners for Mortar Discretization of a Degenerate Evolution Problem

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper we present new efficient variants of substructuring preconditioners for algebraic linear systems arising from the mortar discretization of a degenerate parabolic system of equations. The new approaches extend and adapt the idea of substructuring preconditioners to the discretization of a degenerate problem in electrocardiology. A polylogarithmic bound for the condition number of the preconditioned matrix is proved and validated by numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achdou, Y., Maday, Y., Widlund, O.B.: Iterative substructuring preconditioners for mortar element methods in two dimensions. SIAM J. Numer. Anal. 36(2), 551–580 (1999)

    Article  MathSciNet  Google Scholar 

  2. Bertoluzza, S., Pennacchio, M.: Preconditioning the mortar method by substructuring: the high order case. Appl. Numer. Anal. Comput. Math. 1(3), 434–454 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bertoluzza, S., Pennacchio, M.: Analysis of substructuring preconditioners for mortar methods in an abstract framework. Appl. Math. Lett. 20(2), 131–137 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bertoluzza, S., Perrier, V.: The mortar method in the wavelet context. ESAIM Math. Model. Numer. Anal. 35, 647–673 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructuring I. Math. Comput. 47(175), 103–134 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chan, T.F., Mathew, T.P.: Domain decomposition algorithms. In: Acta Numerica 1994, pp. 61–143. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  7. Cherry, E., Greenside, H., Henriquez, C.H.: Efficient simulation of three-dimensional anisotropic cardiac tissue using an adaptive mesh refinement method. Chaos 3, 853–865 (2003)

    Article  MathSciNet  Google Scholar 

  8. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    Book  MATH  Google Scholar 

  9. Colli Franzone, P., Deuflhard, P., Erdmann, B., Lang, J., Pavarino, L.: Adaptivity in space and time for reaction-diffusion systems in electrocardiology. SIAM J. Sci. Comput. 28, 942–962 (2005)

    Article  MathSciNet  Google Scholar 

  10. Colli Franzone, P., Guerri, L.: Spreading of excitation in 3-D models of the anisotropic cardiac tissue. I: Validation of the eikonal model. Math. Biosci. 113, 145–209 (1993)

    Article  MATH  Google Scholar 

  11. Colli Franzone, P., Pavarino, L.: A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Models Methods Appl. Sci. 14(6), 883–911 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Colli Franzone, P., Savaré, G.: Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level. In: Evolution Equations, Semigroups and Functional Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol. 50, pp. 49–78. Birkhäuser, Basel (2002)

    Google Scholar 

  13. Dryja, M.: Substructuring methods for parabolic problems. In: Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 264–271. SIAM, Philadelphia (1991)

    Google Scholar 

  14. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, London (1985)

    MATH  Google Scholar 

  15. Henriquez, C., Muzikant, A., Smoak, C.: Anisotropy, fiber curvature, and bath loading effects on activation in thin and thick cardiac tissue preparations: Simulations in a three–dimensional bidomain model. J. Cardiovasc. Electrophysiol. 7, 424–444 (1996)

    Article  Google Scholar 

  16. Henriquez, C.S.: Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng. 21, 1–77 (1993)

    Google Scholar 

  17. Hooke, N., Henriquez, C.S., Lanzkron, P., Rose, D.: Linear algebraic transformations of the bidomain equations: implications for numerical methods. Math. Biosci. 120, 127–145 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Keener, J., Bogar, K.: A numerical method for the solution of the bidomain equations in cardiac tissue. Chaos 8, 234–241 (1998)

    Article  MATH  Google Scholar 

  19. Lines, G.T., Buist, M.L., Grøttum, P., Tveito, A.: Modeling the electrical activity of the heart: a bidomain model of the ventricles embedded in a torso. Comput. Vis. Sci. 5(4), 195–213 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)

    Google Scholar 

  21. Luo, C., Rudy, Y.: A model of ventricular cardiac action potential: depolarization, repolarization, and their interaction. Circ. Res. 68, 1501–1526 (1991)

    Google Scholar 

  22. Pennacchio, M.: The mortar finite element method for cardiac reaction-diffusion models. In: Computers in Cardiology, vol. 31, pp. 509–512. IEEE, New York (2004)

    Chapter  Google Scholar 

  23. Pennacchio, M.: The mortar finite element method for the cardiac “bidomain” model of extracellular potential. J. Sci. Comput. 20(2), 191–210 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pennacchio, M.: Substructuring preconditioners for parabolic problems by the mortar method. Int. J. Numer. Anal. Model., to appear

  25. Pennacchio, M., Savaré, G., Colli Franzone, P.: Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37(4), 1333–1370 (2006)

    Article  MATH  Google Scholar 

  26. Pennacchio, M., Simoncini, V.: Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process. J. Comput. Appl. Math. 145(1), 49–70 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pormann, J.B.: A modular simulation system for the bidomain equations. Ph.D. thesis, Dept. of Electrical and Computer Engineering, Duke University (1999)

  28. Smith, B.F., Bjorstad, P.E., Gropp, W.D.: Parallel Multilevel Methods for Partial Differential Equations. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  29. Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34. Springer, New York (2004)

    Google Scholar 

  30. Trangestein, J.A., Kim, C.: Operator splitting and adaptive mesh refinement for the Luo-Rudy I model. J. Comput. Phys. 196, 645–679 (2004)

    Article  MathSciNet  Google Scholar 

  31. Weber Dos Santos, R., Plank, G., Bauer, S., Vigmond, E.: Parallel multigrid preconditioner for the cardiac bidomain model. IEEE Trans. Biomed. Eng. 51, 1960–1968 (2004)

    Article  Google Scholar 

  32. Wohlmuth, B.: Discretization Methods and Iterative Solvers Based on Domain Decomposition. Lecture Notes in Computational Science and Engineering, vol. 17. Springer, Berlin (2001)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto di Matematica Applicata e Tecnologie Informatiche, CNR, via Ferrata, 1, 27100, Pavia, Italy

    Micol Pennacchio

  2. Dipartimento di Matematica and CIRSA, Università di Bologna, Piazza di Porta S. Donato, 5, 40127, Bologna, Italy

    Valeria Simoncini

  3. IMATI-CNR, Pavia, Italy

    Valeria Simoncini

Authors
  1. Micol Pennacchio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valeria Simoncini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Micol Pennacchio.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pennacchio, M., Simoncini, V. Substructuring Preconditioners for Mortar Discretization of a Degenerate Evolution Problem. J Sci Comput 36, 391–419 (2008). https://doi.org/10.1007/s10915-008-9195-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-008-9195-7

Keywords

Search

Navigation