Maximal 𝐿^{𝑝} analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra

Li, Buyang; Sun, Weiwei

doi:10.1090/mcom/3133

Maximal $L^p$ analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra
HTML articles powered by AMS MathViewer

by Buyang Li and Weiwei Sun PDF

Math. Comp. 86 (2017), 1071-1102 Request permission

Abstract:

The paper is concerned with Galerkin finite element solutions of parabolic equations in a convex polygon or polyhedron with a diffusion coefficient in $W^{1,N+\alpha }$ for some $\alpha >0$, where $N$ denotes the dimension of the domain. We prove the analyticity of the semigroup generated by the discrete elliptic operator, the discrete maximal $L^p$ regularity and the optimal $L^p$ error estimate of the finite element solution for the parabolic equation.

References

Nikolai Yu. Bakaev, Maximum norm resolvent estimates for elliptic finite element operators, BIT 41 (2001), no. 2, 215–239. MR 1837395, DOI 10.1023/A:1021934205234
Nikolai Yu. Bakaev, Vidar Thomée, and Lars B. Wahlbin, Maximum-norm estimates for resolvents of elliptic finite element operators, Math. Comp. 72 (2003), no. 244, 1597–1610. MR 1986795, DOI 10.1090/S0025-5718-02-01488-6
P. Chatzipantelidis, R. D. Lazarov, V. Thomée, and L. B. Wahlbin, Parabolic finite element equations in nonconvex polygonal domains, BIT 46 (2006), no. suppl., S113–S143. MR 2283311, DOI 10.1007/s10543-006-0087-7
H. Chen, An $L^2$ and $L^\infty$-error analysis for parabolic finite element equations with applications by superconvergence and error expansions, Doctoral Dissertation, Heidelberg University, 1993.
Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
Michel Crouzeix and Vidar Thomée, Resolvent estimates in $l_p$ for discrete Laplacians on irregular meshes and maximum-norm stability of parabolic finite difference schemes, Comput. Methods Appl. Math. 1 (2001), no. 1, 3–17. MR 1839793, DOI 10.2478/cmam-2001-0001
A. Demlow, D. Leykekhman, A. H. Schatz, and L. B. Wahlbin, Best approximation property in the $W^{1}_{\infty }$ norm for finite element methods on graded meshes, Math. Comp. 81 (2012), no. 278, 743–764. MR 2869035, DOI 10.1090/S0025-5718-2011-02546-9
Alan Demlow and Charalambos Makridakis, Sharply local pointwise a posteriori error estimates for parabolic problems, Math. Comp. 79 (2010), no. 271, 1233–1262. MR 2629992, DOI 10.1090/S0025-5718-10-02346-X
Jim Douglas Jr., The numerical simulation of miscible displacement in porous media, Computational methods in nonlinear mechanics (Proc. Second Internat. Conf., Univ. Texas, Austin, Tex., 1979) North-Holland, Amsterdam-New York, 1980, pp. 225–237. MR 576907
S. D. Èĭdel′man and S. D. Ivasišen, Investigation of the Green’s matrix of a homogeneous parabolic boundary value problem, Trudy Moskov. Mat. Obšč. 23 (1970), 179–234 (Russian). MR 0367455
Richard E. Ewing, Yanping Lin, Junping Wang, and Shuhua Zhang, $L^\infty$-error estimates and superconvergence in maximum norm of mixed finite element methods for non-Fickian flows in porous media, Int. J. Numer. Anal. Model. 2 (2005), no. 3, 301–328. MR 2112650
E. B. Fabes and D. W. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), no. 4, 327–338. MR 855753, DOI 10.1007/BF00251802
Stephen J. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc. 119 (1993), no. 1, 225–233. MR 1156467, DOI 10.1090/S0002-9939-1993-1156467-3
Matthias Geissert, Discrete maximal $L_p$ regularity for finite element operators, SIAM J. Numer. Anal. 44 (2006), no. 2, 677–698. MR 2218965, DOI 10.1137/040616553
Matthias Geissert, Applications of discrete maximal $L_p$ regularity for finite element operators, Numer. Math. 108 (2007), no. 1, 121–149. MR 2350187, DOI 10.1007/s00211-007-0110-1
Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
Pierre Grisvard, Elliptic problems in nonsmooth domains, Classics in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Reprint of the 1985 original [ MR0775683]; With a foreword by Susanne C. Brenner. MR 3396210, DOI 10.1137/1.9781611972030.ch1
Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), no. 3, 303–342. MR 657523, DOI 10.1007/BF01166225
J. Guzmán, D. Leykekhman, J. Rossmann, and A. H. Schatz, Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods, Numer. Math. 112 (2009), no. 2, 221–243. MR 2495783, DOI 10.1007/s00211-009-0213-y
Anita Hansbo, Strong stability and non-smooth data error estimates for discretizations of linear parabolic problems, BIT 42 (2002), no. 2, 351–379. MR 1912592, DOI 10.1023/A:1021903109720
Huilian Jia, Dongsheng Li, and Lihe Wang, Global regularity for divergence form elliptic equations on quasiconvex domains, J. Differential Equations 249 (2010), no. 12, 3132–3147. MR 2737424, DOI 10.1016/j.jde.2010.08.015
B. Kovács, B. Li, and Ch. Lubich, $A$-stable time discretizations preserve maximal parabolic regularity, http://arxiv.org/abs/1511.07823
Peer C. Kunstmann and Lutz Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65–311. MR 2108959, DOI 10.1007/978-3-540-44653-8_{2}
Dmitriy Leykekhman, Pointwise localized error estimates for parabolic finite element equations, Numer. Math. 96 (2004), no. 3, 583–600. MR 2028727, DOI 10.1007/s00211-003-0480-y
Buyang Li, Maximum-norm stability and maximal $L^p$ regularity of FEMs for parabolic equations with Lipschitz continuous coefficients, Numer. Math. 131 (2015), no. 3, 489–516. MR 3395142, DOI 10.1007/s00211-015-0698-5
Buyang Li and Weiwei Sun, Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media, SIAM J. Numer. Anal. 51 (2013), no. 4, 1959–1977. MR 3072763, DOI 10.1137/120871821
Buyang Li and Weiwei Sun, Regularity of the diffusion-dispersion tensor and error analysis of Galerkin FEMs for a porous medium flow, SIAM J. Numer. Anal. 53 (2015), no. 3, 1418–1437. MR 3355773, DOI 10.1137/140958803
Y. P. Lin, On maximum norm estimates for Ritz-Volterra projection with applications to some time dependent problems, J. Comput. Math. 15 (1997), no. 2, 159–178. MR 1448820
Yan Ping Lin, Vidar Thomée, and Lars B. Wahlbin, Ritz-Volterra projections to finite-element spaces and applications to integrodifferential and related equations, SIAM J. Numer. Anal. 28 (1991), no. 4, 1047–1070. MR 1111453, DOI 10.1137/0728056
Christian Lubich and Olavi Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), no. 2, 293–313. MR 1112225, DOI 10.1007/BF01931289
Timothy A. Lucas, Maximum-norm estimates for an immunology model using reaction-diffusion equations with stochastic source terms, SIAM J. Numer. Anal. 49 (2011), no. 6, 2256–2276. MR 2854595, DOI 10.1137/100794584
J. A. Nitsche and Mary F. Wheeler, $L_{\infty }$-boundedness of the finite element Galerkin operator for parabolic problems, Numer. Funct. Anal. Optim. 4 (1981/82), no. 4, 325–353. MR 673316, DOI 10.1080/01630568208816121
El-Maati Ouhabaz, Gaussian estimates and holomorphy of semigroups, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1465–1474. MR 1232142, DOI 10.1090/S0002-9939-1995-1232142-3
C. Palencia, Maximum norm analysis of completely discrete finite element methods for parabolic problems, SIAM J. Numer. Anal. 33 (1996), no. 4, 1654–1668. MR 1403564, DOI 10.1137/S0036142993259779
Rolf Rannacher, $L^\infty$-stability estimates and asymptotic error expansion for parabolic finite element equations, Extrapolation and defect correction (1990), Bonner Math. Schriften, vol. 228, Univ. Bonn, Bonn, 1991, pp. 74–94. MR 1185533
Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437–445. MR 645661, DOI 10.1090/S0025-5718-1982-0645661-4
Alfred H. Schatz, A weak discrete maximum principle and stability of the finite element method in $L_{\infty }$ on plane polygonal domains. I, Math. Comp. 34 (1980), no. 149, 77–91. MR 551291, DOI 10.1090/S0025-5718-1980-0551291-3
A. H. Schatz, V. C. Thomée, and L. B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl. Math. 33 (1980), no. 3, 265–304. MR 562737, DOI 10.1002/cpa.3160330305
A. H. Schatz, V. Thomée, and L. B. Wahlbin, Stability, analyticity, and almost best approximation in maximum norm for parabolic finite element equations, Comm. Pure Appl. Math. 51 (1998), no. 11-12, 1349–1385. MR 1639143, DOI 10.1002/(SICI)1097-0312(199811/12)51:11/12<1349::AID-CPA5>3.3.CO;2-T
A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), no. 138, 414–442. MR 431753, DOI 10.1090/S0025-5718-1977-0431753-X
A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods. II, Math. Comp. 64 (1995), no. 211, 907–928. MR 1297478, DOI 10.1090/S0025-5718-1995-1297478-7
Zhongwei Shen, Bounds of Riesz transforms on $L^p$ spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 173–197 (English, with English and French summaries). MR 2141694
Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
Vidar Thomée, Maximum-norm stability, smoothing and resolvent estimates for parabolic finite element equations, ESAIM Proceedings. Vol. 21 (2007) [Journées d’Analyse Fonctionnelle et Numérique en l’honneur de Michel Crouzeix], ESAIM Proc., vol. 21, EDP Sci., Les Ulis, 2007, pp. 98–107 (English, with English and French summaries). MR 2404056, DOI 10.1051/proc:072108
Vidar Thomée, Galerkin finite element methods for parabolic problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR 2249024
Vidar Thomée and Lars B. Wahlbin, Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable, Numer. Math. 41 (1983), no. 3, 345–371. MR 712117, DOI 10.1007/BF01418330
V. Thomée and L. B. Wahlbin, Stability and analyticity in maximum-norm for simplicial Lagrange finite element semidiscretizations of parabolic equations with Dirichlet boundary conditions, Numer. Math. 87 (2000), no. 2, 373–389. MR 1804662, DOI 10.1007/s002110000184
Lars B. Wahlbin, A quasioptimal estimate in piecewise polynomial Galerkin approximation of parabolic problems, Numerical analysis (Dundee, 1981) Lecture Notes in Math., vol. 912, Springer, Berlin-New York, 1982, pp. 230–245. MR 654353
Lutz Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math. Ann. 319 (2001), no. 4, 735–758. MR 1825406, DOI 10.1007/PL00004457
Lutz Weis, A new approach to maximal $L_p$-regularity, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998) Lecture Notes in Pure and Appl. Math., vol. 215, Dekker, New York, 2001, pp. 195–214. MR 1818002