Skip to main content
SpringerLink
Log in

Discrete maximal parabolic regularity for Galerkin finite element methods

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the time discontinuous Galerkin solution of linear parabolic equations. Such estimates have many applications. They are essential, for example, in establishing optimal a priori error estimates in non-Hilbertian norms without unnatural coupling of spatial mesh sizes with time steps.

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. Coulhon, T., Duong, X.T.: Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss. Adv. Differ. Equ. 5(1–3), 343–368 (2000)

    MATH  Google Scholar 

  2. Haller-Dintelmann, R., Rehberg, J.: Maximal parabolic regularity for divergence operators including mixed boundary conditions. J. Differ. Equ. 247(5), 1354–1396 (2009). doi:10.1016/j.jde.2009.06.001

    Article  MathSciNet  MATH  Google Scholar 

  3. Hieber, M., Prüss, J.: Heat kernels and maximal \(L^p\)-\(L^q\) estimates for parabolic evolution equations. Commun. Partial Differ. Equ. 22(9–10), 1647–1669 (1997). doi:10.1080/03605309708821314

    MathSciNet  MATH  Google Scholar 

  4. Hömberg, D., Meyer, C., Rehberg, J., Ring, W.: Optimal control for the thermistor problem. SIAM J. Control Optim. 48(5), 3449–3481 (2009/10). doi:10.1137/080736259

  5. Krumbiegel, K., Rehberg, J.: Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints. SIAM J. Control Optim. 51(1), 304–331 (2013). doi:10.1137/120871687

    Article  MathSciNet  MATH  Google Scholar 

  6. Kunisch, K., Pieper, K., Vexler, B.: Measure valued directional sparsity for parabolic optimal control problems. SIAM J. Control Optim. 52(5), 3078–3108 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Leykekhman, D., Vexler, B.: Optimal a priori error estimates of parabolic optimal control problems with pointwise control. SIAM J. Numer. Anal. 51(5), 2797–2821 (2013). doi:10.1137/120885772

    Article  MathSciNet  MATH  Google Scholar 

  8. Leykekhman, D., Vexler, B.: A priori error estimates for three dimensional parabolic optimal control problems with pointwise control. SIAM J. Control Optim. (2016) (Accepted)

  9. Leykekhman, D., Vexler, B.: Pointwise best approximation results for Galerkin finite element solutions of parabolic problems. SIAM J. Numer. Anal. 54(3), 1365–1384 (2016). doi:10.1137/15M103412X

    Article  MathSciNet  MATH  Google Scholar 

  10. Geissert, M.: Discrete maximal \(L_p\) regularity for finite element operators. SIAM J. Numer. Anal. 44(2), 677–698 (electronic) (2006). doi:10.1137/040616553

  11. Geissert, M.: Applications of discrete maximal \(L_p\) regularity for finite element operators. Numer. Math. 108(1), 121–149 (2007). doi:10.1007/s00211-007-0110-1

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, B.: Maximum-norm stability and maximal \(L^p\) regularity of FEMs for parabolic equations with Lipschitz continuous coefficients. Numer. Math. 131(3), 489–516 (2015)

  13. Blunck, S.: Analyticity and discrete maximal regularity on \(L_p\)-spaces. J. Funct. Anal. 183(1), 211–230 (2001). doi:10.1006/jfan.2001.3740

    Article  MathSciNet  MATH  Google Scholar 

  14. Blunck, S.: Maximal regularity of discrete and continuous time evolution equations. Stud. Math. 146(2), 157–176 (2001). doi:10.4064/sm146-2-3

    Article  MathSciNet  MATH  Google Scholar 

  15. Portal, P.: Maximal regularity of evolution equations on discrete time scales. J. Math. Anal. Appl. 304(1), 1–12 (2005). doi:10.1016/j.jmaa.2004.09.003

    Article  MathSciNet  MATH  Google Scholar 

  16. Guidetti, D.: Backward Euler scheme, singular Hölder norms, and maximal regularity for parabolic difference equations. Numer. Funct. Anal. Optim. 28(3–4), 307–337 (2007). doi:10.1080/01630560701285594

    Article  MathSciNet  MATH  Google Scholar 

  17. Guidetti, D.: Maximal regularity for parabolic equations and implicit Euler scheme. In: Mathematical analysis seminar, University of Bologna Department of Mathematics: Academic Year 2003/2004. Academic Year 2004/2005 (Italian), pp. 253–264. Tecnoprint, Bologna (2007)

  18. Ashyralyev, A., Sobolevskiĭ, P.E.: Well-posedness of parabolic difference equations. In: Operator Theory: Advances and Applications, vol. 69. Birkhäuser Verlag, Basel (1994). doi:10.1007/978-3-0348-8518-8 (Translated from the Russian by A. Iacob)

  19. Eriksson, K., Johnson, C., Thomée, V.: Time discretization of parabolic problems by the discontinuous Galerkin method. RAIRO Modél. Math. Anal. Numér. 19(4), 611–643 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems. I. A linear model problem. SIAM J. Numer. Anal. 28(1), 43–77 (1991). doi:10.1137/0728003

    Article  MathSciNet  MATH  Google Scholar 

  21. Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems. II. Optimal error estimates in \(L_\infty L_2\) and \(L_\infty L_\infty \). SIAM J. Numer. Anal. 32(3), 706–740 (1995). doi:10.1137/0732033

    Article  MathSciNet  MATH  Google Scholar 

  22. Schmich, M., Vexler, B.: Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations. SIAM J. Sci. Comput. 30(1), 369–393 (2007/08). doi:10.1137/060670468

  23. Schötzau, D., Wihler, T.P.: A posteriori error estimation for \(hp\)-version time-stepping methods for parabolic partial differential equations. Numer. Math. 115(3), 475–509 (2010). doi:10.1007/s00211-009-0285-8

    Article  MathSciNet  MATH  Google Scholar 

  24. Lasaint, P., Raviart, P.A.: On a finite element method for solving the neutron transport equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), pp. 89–123. Publication No. 33. Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York (1974)

  25. Richter, T., Springer, A., Vexler, B.: Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems. Numer. Math. 124(1), 151–182 (2013). doi:10.1007/s00211-012-0511-7

    Article  MathSciNet  MATH  Google Scholar 

  26. Becker, R., Meidner, D., Vexler, B.: Efficient numerical solution of parabolic optimization problems by finite element methods. Optim. Methods Softw. 22(5), 813–833 (2007). doi:10.1080/10556780701228532

    Article  MathSciNet  MATH  Google Scholar 

  27. Meidner, D., Vexler, B.: A priori error analysis of the Petrov–Galerkin Crank–Nicolson scheme for parabolic optimal control problems. SIAM J. Control Optim. 49(5), 2183–2211 (2011). doi:10.1137/100809611

    Article  MathSciNet  MATH  Google Scholar 

  28. Chrysafinos, K.: Discontinuous Galerkin approximations for distributed optimal control problems constrained by parabolic PDE’s. Int. J. Numer. Anal. Model. 4(3–4), 690–712 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Meidner, D., Vexler, B.: Adaptive space-time finite element methods for parabolic optimization problems. SIAM J. Control Optim. 46(1), 116–142 (electronic) (2007). doi:10.1137/060648994

  30. Meidner, D., Vexler, B.: A priori error estimates for space-time finite element approximation of parabolic optimal control problems. Part I: problems without control constraints. SIAM J. Control Optim. 47(3), 1150–1177 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Meidner, D., Vexler, B.: A priori error estimates for space-time finite element approximation of parabolic optimal control problems. Part II: problems with control constraints. SIAM J. Control Optim. 47(3), 1301–1329 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Meidner, D., Rannacher, R., Vexler, B.: A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time. SIAM J. Control Optim. 49(5), 1961–1997 (2011). doi:10.1137/100793888

    Article  MathSciNet  MATH  Google Scholar 

  33. Thomée, V.: Galerkin finite element methods for parabolic problems. Springer Series in Computational Mathematics, vol. 25, 2nd edn. Springer, Berlin (2006)

  34. Shen, Z.W.: Resolvent estimates in \(L^p\) for elliptic systems in Lipschitz domains. J. Funct. Anal. 133(1), 224–251 (1995). doi:10.1006/jfan.1995.1124

    Article  MathSciNet  MATH  Google Scholar 

  35. Haller-Dintelmann, R., Meyer, C., Rehberg, J., Schiela, A.: Hölder continuity and optimal control for nonsmooth elliptic problems. Appl. Math. Optim. 60(3), 397–428 (2009). doi:10.1007/s00245-009-9077-x

    Article  MathSciNet  MATH  Google Scholar 

  36. Eriksson, K., Johnson, C., Larsson, S.: Adaptive finite element methods for parabolic problems. VI. Analytic semigroups. SIAM J. Numer. Anal. 35(4), 1315–1325 (electronic) (1998). doi:10.1137/S0036142996310216

  37. Egert, M., Rozendaal, J.: Convergence of subdiagonal Padé approximations of \(C_0\)-semigroups. J. Evol. Equ. 13(4), 875–895 (2013). doi:10.1007/s00028-013-0207-1

    Article  MathSciNet  MATH  Google Scholar 

  38. Bakaev, N.Y., Thomée, V., Wahlbin, L.B.: Maximum-norm estimates for resolvents of elliptic finite element operators. Math. Comput. 72(244), 1597–1610 (electronic) (2003). doi:10.1090/S0025-5718-02-01488-6

  39. Li, B., Sun, W.: Maximal \(l^p\) analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra. (Preprint) arXiv:1501.07345 (2015)

  40. Leykekhman, D., Vexler, B.: Finite element pointwise results on convex polyhedral domains. SIAM J. Numer. Anal. 54(2), 561–587 (2016). doi:10.1137/15M1013912

    Article  MathSciNet  MATH  Google Scholar 

  41. Bakaev, N.Y., Crouzeix, M., Thomée, V.: Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations. M2AN Math. Model. Numer. Anal. 40(5), 923–937 (2006). doi:10.1051/m2an:2006040 (2007)

Download references

Acknowledgments

The authors would like to thank Dominik Meidner and Konstantin Pieper for the careful reading of the manuscript and providing valuable suggestions that help to improve the presentation of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Dmitriy Leykekhman

  2. Lehrstuhl für Optimalsteuerung, Fakultät für Mathematik, Technische Universität München, 85748, Boltzmannstraße 3, Garching b. München, Germany

    Boris Vexler

Authors
  1. Dmitriy Leykekhman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris Vexler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris Vexler.

Additional information

The author was partially supported by NSF Grant DMS-1522555.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leykekhman, D., Vexler, B. Discrete maximal parabolic regularity for Galerkin finite element methods. Numer. Math. 135, 923–952 (2017). https://doi.org/10.1007/s00211-016-0821-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-016-0821-2

Search

Navigation