Skip to main content
SpringerLink
Log in

Applications of discrete maximal L p regularity for finite element operators

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we present applications of discrete maximal L p regularity for finite element operators. More precisely, we show error estimates of order h 2 for linear and certain semilinear problems in various L p (Ω)-norms. Discrete maximal regularity allows us to prove error estimates in a very easy and efficient way. Moreover, we also develop interpolation theory for (fractional powers of) finite element operators and extend the results on discrete maximal L p regularity formerly proved by the author.

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. Arendt W. (1994). Gaussian estimates and interpolation of the spectrum in L p. Diff. Integr. Equ. 7(5–6): 1153–1168

    MATH  MathSciNet  Google Scholar 

  2. Arendt W. and ter Elst A.F.M. (1997). Gaussian estimates for second order elliptic operators with boundary conditions. J. Oper. Theory 38(1): 87–130

    MATH  MathSciNet  Google Scholar 

  3. Bakaev N.Y. (2001). Maximum norm resolvent estimates for elliptic finite element operators. BIT 41(2): 215–239

    Article  MATH  MathSciNet  Google Scholar 

  4. Bakaev N.Y., Thomée V. and Wahlbin L.B. (2003). Maximum-norm estimates for resolvents of elliptic finite element operators. Math. Comp. 72(244): 1597–1610 (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  5. Denk R., Hieber M. and Prüss J. (2003). \({\mathcal{R}}\) -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166(788): viii+114

    Google Scholar 

  6. Dore, G.: L p regularity for abstract differential equations. In: Functional Analysis and Related Topics, 1991 (Kyoto), pp. 25–38. Springer, Berlin (1993)

  7. Èĭ del’man S.D. and Ivasišen S.D. (1970). Investigation of the Green’s matrix of a homogeneous parabolic boundary value problem. Trudy Moskov. Mat. Obšč. 23: 179–234

    Google Scholar 

  8. Geissert, M.: Maximal L p -L q Regularity for discrete elliptic differential operators. PhD thesis, TU Darmstadt, July (2003)

  9. Geissert M. (2006). Discrete maximal L p regularity for finite element operators. SIAM J. Numer. Anal. 44(2): 677–698 (electronic)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  11. Lunardi A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser Verlag, Basel

    MATH  Google Scholar 

  12. Palencia C. (1996). Maximum norm analysis of completely discrete finite element methods for parabolic problems. SIAM J. Numer. Anal. 33(4): 1654–1668

    Article  MATH  MathSciNet  Google Scholar 

  13. Pazy A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York

    MATH  Google Scholar 

  14. Rannacher R. and Scott R. (1982). Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38(158): 437–445

    Article  MATH  MathSciNet  Google Scholar 

  15. Schatz A.H., Thomée V. and Wahlbin L.B. (1998). Stability, analyticity, and almost best approximation in maximum norm for parabolic finite element equations. Commun. Pure Appl. Math. 51(11–12): 1349–1385

    Article  MATH  Google Scholar 

  16. Schatz A.H. and Wahlbin L.B. (1982). On the quasi-optimality in L ∞ of the Ḣ1-projection into finite element spaces. Math. Comp. 38(157): 1–22

    Article  MATH  MathSciNet  Google Scholar 

  17. Thomée V. and Wahlbin L.B. (2000). Stability and analyticity in maximum-norm for simplicial Lagrange finite element semidiscretizations of parabolic equations with Dirichlet boundary conditions. Numer. Math. 87(2): 373–389

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Matthias Geissert

Authors
  1. Matthias Geissert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Geissert.

Additional information

The author was supported by the DFG-Graduiertenkolleg 853.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-007-0110-1

Mathematics Subject Classification (2000)

Search

Navigation