Stability, analyticity, and maximal regularity for parabolic finite element problems on smooth domains

Kashiwabara, Takahito; Kemmochi, Tomoya

doi:10.1090/mcom/3500

Stability, analyticity, and maximal regularity for parabolic finite element problems on smooth domains
HTML articles powered by AMS MathViewer

by Takahito Kashiwabara and Tomoya Kemmochi HTML | PDF

Math. Comp. 89 (2020), 1647-1679 Request permission

Abstract:

In this paper, we consider the finite element semidiscretization for a parabolic problem on a smooth domain $\Omega \subset \mathbb {R}^N$ with the Neumann boundary condition. We emphasize that the domain can be nonconvex in general. We discretize the this problem by the finite element method by constructing a family of polygonal or polyhedral domains $\{ \Omega _h \}_h$ that approximate the original domain $\Omega$. The aim of this study is to derive the smoothing property for the discrete parabolic semigroup and the maximal regularity for the discrete elliptic operator. The main difficulty is the effect of the boundary-skin (symmetric difference) $\Omega \bigtriangleup \Omega _h$. In order to address the effect of the boundary-skin, we introduce the tubular neighborhood of the original boundary $\partial \Omega$.

References

M. S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N. Wells, The fenics project version 1.5, Archive of Numerical Software 3 (2015), no. 100.
John W. Barrett and Charles M. Elliott, Finite-element approximation of elliptic equations with a Neumann or Robin condition on a curved boundary, IMA J. Numer. Anal. 8 (1988), no. 3, 321–342. MR 968097, DOI 10.1093/imanum/8.3.321
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Wentao Cai, Buyang Li, Yanping Lin, and Weiwei Sun, Analysis of fully discrete FEM for miscible displacement in porous media with Bear-Scheidegger diffusion tensor, Numer. Math. 141 (2019), no. 4, 1009–1042. MR 3923520, DOI 10.1007/s00211-019-01030-0
Calixto P. Calderón and Mario Milman, Interpolation of Sobolev spaces. The real method, Indiana Univ. Math. J. 32 (1983), no. 6, 801–808. MR 721564, DOI 10.1512/iumj.1983.32.32054
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
Giovanni Dore, $L^p$ regularity for abstract differential equations, Functional analysis and related topics, 1991 (Kyoto), Lecture Notes in Math., vol. 1540, Springer, Berlin, 1993, pp. 25–38. MR 1225809, DOI 10.1007/BFb0085472
Kenneth Eriksson, Claes Johnson, and Stig Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups, SIAM J. Numer. Anal. 35 (1998), no. 4, 1315–1325. MR 1620144, DOI 10.1137/S0036142996310216
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
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640, DOI 10.1515/jnum-2012-0013
T. Kashiwabara and T. Kemmochi, Pointwise error estimates of linear finite element method for Neumann boundary value problems in a smooth domain, Numer. Math. (published electronically January 4, 2020). DOI 10.1007/s00211-019-01098-8.
Takahito Kashiwabara, Issei Oikawa, and Guanyu Zhou, Penalty method with P1/P1 finite element approximation for the Stokes equations under the slip boundary condition, Numer. Math. 134 (2016), no. 4, 705–740. MR 3563279, DOI 10.1007/s00211-016-0790-5
Tomoya Kemmochi and Norikazu Saito, Discrete maximal regularity and the finite element method for parabolic equations, Numer. Math. 138 (2018), no. 4, 905–937. MR 3778340, DOI 10.1007/s00211-017-0929-z
Dmitriy Leykekhman and Boris Vexler, Pointwise best approximation results for Galerkin finite element solutions of parabolic problems, SIAM J. Numer. Anal. 54 (2016), no. 3, 1365–1384. MR 3498514, DOI 10.1137/15M103412X
Dmitriy Leykekhman and Boris Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods, Numer. Math. 135 (2017), no. 3, 923–952. MR 3606467, DOI 10.1007/s00211-016-0821-2
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, 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
Norikazu Saito, Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis, Commun. Pure Appl. Anal. 11 (2012), no. 1, 339–364. MR 2833351, DOI 10.3934/cpaa.2012.11.339
A. H. Schatz, I. H. Sloan, and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. 33 (1996), no. 2, 505–521. MR 1388486, DOI 10.1137/0733027
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
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
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, Local behavior in finite element methods, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 353–522. MR 1115238
Guanyu Zhou and Norikazu Saito, Finite volume methods for a Keller-Segel system: discrete energy, error estimates and numerical blow-up analysis, Numer. Math. 135 (2017), no. 1, 265–311. MR 3592113, DOI 10.1007/s00211-016-0793-2