Advertisement

Foundations of Computational Mathematics

, Volume 15, Issue 6, pp 1653–1701 | Cite as

On a Full Discretisation for Nonlinear Second-Order Evolution Equations with Monotone Damping: Construction, Convergence, and Error Estimates

  • Etienne EmmrichEmail author
  • David Šiška
  • Mechthild Thalhammer
Article

Abstract

Convergence of a full discretisation method is studied for a class of nonlinear second order in time evolution equations, where the nonlinear operator acting on the first-order time derivative of the solution is supposed to be hemicontinuous, monotone, coercive and to satisfy a certain growth condition, and the operator acting on the solution is assumed to be linear, bounded, symmetric, and strongly positive. The numerical approximation combines a Galerkin spatial discretisation with a novel time discretisation obtained from a reformulation of the second-order evolution equation as a first-order system and an application of the two-step backward differentiation formula with constant time stepsizes. Convergence towards the weak solution is shown for suitably chosen piecewise polynomial in time prolongations of the resulting fully discrete solutions, and an a priori error estimate ensures convergence of second order in time provided that the exact solution to the problem fulfils certain regularity requirements. A numerical example for a model problem describing the displacement of a vibrating membrane in a viscous medium illustrates the favourable error behaviour of the proposed full discretisation method in situations where regular solutions exist.

Keywords

Nonlinear evolution equation of second order in time Monotone operator Weak solution Time discretisation Convergence 

Mathematics Subject Classification

65M12 47J35 35G25 34G20 47H05 

Notes

Acknowledgments

We acknowledge financial support by the SFB 910 under project A8, DFG.

References

  1. 1.
    G. Andreassi and G. Torelli, Su una equazione di tipo iperbolico non lineare, Rend. Sem. Mat. Univ. Padova 35/1 (1965) 134–147.Google Scholar
  2. 2.
    R. E. Bank and H. Yserentant, On the \(H^1\)-stability of the \(L_2\)-projection onto finite element spaces, Numer. Math. 126/2 (2014) 361–381.Google Scholar
  3. 3.
    V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff Int. Publ., Leyden, 1976.Google Scholar
  4. 4.
    M. Boman, Estimates for the \(L_2\)-projection onto continuous finite element spaces in a weighted \(L_p\)-norm, BIT Numer. Math. 46 (2006) 249–260.Google Scholar
  5. 5.
    H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2010.Google Scholar
  6. 6.
    C. Carstensen, Merging the Bramble–Pasciak–Steinbach and the Crouzeix–Thomée criterion for \(H^1\)-stability of the \(L^2\)-projection onto finite element spaces, Math. Comp. 71 (2002) 157–163.Google Scholar
  7. 7.
    P. Ciarlet, Basic error estimates for elliptic problems, in P. Ciarlet and J.-L. Lions (eds.), Handbook of Numerical Analysis II (Part 1), Elsevier, Amsterdam, 1998.Google Scholar
  8. 8.
    B. Cockburn, On the continuity in \(\text{ BV }(\varOmega )\) of the \(L^2\)-projection into finite element spaces, Math. Comp. 57/196 (1991) 551–561.Google Scholar
  9. 9.
    P. Colli and A. Favini, Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations, Internat. J. Math. Math. Sci. 19/3 (1996) 481–494.Google Scholar
  10. 10.
    M. Crouzeix and V. Thomée, The stability in \(L_p\) and \(W^1_p\) of the \(L_2\)-projection onto finite element function spaces, Math. Comp. 48/178 (1987) 521–532.Google Scholar
  11. 11.
    J. Diestel and J. Uhl, Vector measures, American Mathematical Society, Providence, R.I., 1977.Google Scholar
  12. 12.
    R. Edwards, Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, 1965.Google Scholar
  13. 13.
    E. Emmrich, Stability and convergence of the two-step BDF for the incompressible Navier-Stokes problem, Int. J. Nonlinear Sci. Numer. Simul. 5/3 (2004) 199–210.Google Scholar
  14. 14.
    E. Emmrich, Convergence of a time discretization for a class of non-Newtonian fluid flow, Commun. Math. Sci. 6/4 (2008) 827–843.Google Scholar
  15. 15.
    E. Emmrich, Two-step BDF time discretisation of nonlinear evolution problems governed by monotone operators with strongly continuous perturbations, Comput. Methods Appl. Math. 9/1 (2009) 37–62.Google Scholar
  16. 16.
    E. Emmrich and D. Šiška, Full discretization of second-order nonlinear evolution equations: strong convergence and applications, Comput. Methods Appl. Math. 11/4 (2011) 441–459.Google Scholar
  17. 17.
    E. Emmrich and D. Šiška, Full discretization of the porous medium/fast diffusion equation based on its very weak formulation, Commun. Math. Sci. 10/4 (2012) 1055–1080.Google Scholar
  18. 18.
    E. Emmrich and M. Thalhammer, Stiffly accurate Runge-Kutta methods for nonlinear evolution problems governed by a monotone operator, Math. Comp. 79 (2010) 785–806.Google Scholar
  19. 19.
    E. Emmrich and M. Thalhammer, Convergence of a time discretisation for doubly nonlinear evolution equations of second order, Found. Comput. Math. 10/2 (2010) 171–190.Google Scholar
  20. 20.
    E. Emmrich and M. Thalhammer, Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation, J. Differential Equations 251 (2011) 82–118.Google Scholar
  21. 21.
    A. Friedman and J. Nečas, Systems of nonlinear wave equations with nonlinear viscosity, Pac. J. Math. 135/1 (1988) 29–55.Google Scholar
  22. 22.
    H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.Google Scholar
  23. 23.
    C. Levermore, T. Manteuffel, and A. White, Numerical solutions of partial differential equations on irregular grids, in Computational Techniques and Applications CTAC-87, Sydney, 1987, 417–426, North-Holland, Amsterdam/New York, 1987.Google Scholar
  24. 24.
    J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.Google Scholar
  25. 25.
    J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Vol. 1, Dunod, Gauthier-Villars, Paris, 1968.Google Scholar
  26. 26.
    J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France 93 (1965) 43–96.Google Scholar
  27. 27.
    T. Manteuffel and A. White, The numerical solution of second-order boundary value problems on nonuniform meshes, Math. Comp. 47/176 (1986) 511–535.Google Scholar
  28. 28.
    T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2005.Google Scholar
  29. 29.
    O. Steinbach, On the stability of the \(L_2\) projection in fractional Sobolev spaces, Numer. Math. 88 (2001) 367–379.Google Scholar
  30. 30.
    R. Temam, Numerical Analysis, D. Reidel, Dordrecht, 1973.Google Scholar
  31. 31.
    E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators, Springer, New York, 1990.Google Scholar

Copyright information

© SFoCM 2014

Authors and Affiliations

  • Etienne Emmrich
    • 1
    Email author
  • David Šiška
    • 2
  • Mechthild Thalhammer
    • 3
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.School of Mathematical SciencesUniversity of EdinburghEdinburghUK
  3. 3.Institut für MathematikLeopold-Franzens-Universität InnsbruckInnsbruckAustria

Personalised recommendations