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.
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Arendt W. (1994). Gaussian estimates and interpolation of the spectrum in L p. Diff. Integr. Equ. 7(5–6): 1153–1168
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
Bakaev N.Y. (2001). Maximum norm resolvent estimates for elliptic finite element operators. BIT 41(2): 215–239
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)
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
Dore, G.: L p regularity for abstract differential equations. In: Functional Analysis and Related Topics, 1991 (Kyoto), pp. 25–38. Springer, Berlin (1993)
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
Geissert, M.: Maximal L p -L q Regularity for discrete elliptic differential operators. PhD thesis, TU Darmstadt, July (2003)
Geissert M. (2006). Discrete maximal L p regularity for finite element operators. SIAM J. Numer. Anal. 44(2): 677–698 (electronic)
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
Lunardi A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser Verlag, Basel
Palencia C. (1996). Maximum norm analysis of completely discrete finite element methods for parabolic problems. SIAM J. Numer. Anal. 33(4): 1654–1668
Pazy A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York
Rannacher R. and Scott R. (1982). Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38(158): 437–445
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
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
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
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The author was supported by the DFG-Graduiertenkolleg 853.
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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
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DOI: https://doi.org/10.1007/s00211-007-0110-1