Discrete Maximal Regularity for the Discontinuous Galerkin Time-Stepping Method without Logarithmic Factor
Abstract.
Maximal regularity is a kind of a priori estimate for parabolic-type equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green’s function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.
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References
1.
R. P. Agarwal, C. Cuevas, and C. Lizama, Regularity of Difference Equations on Banach Spaces, Springer, Cham, 2014.
2.
G. Akrivis and B. Li, Linearization of the finite element method for gradient flows by Newton’s method, IMA J. Numer. Anal., 41 (2021), pp. 1411–1440.
3.
G. Akrivis and C. Makridakis, On maximal regularity estimates for discontinuous Galerkin time-discrete methods, SIAM J. Numer. Anal., 60 (2022), pp. 180–194, https://doi.org/10.1137/20M1383781.
4.
A. Ashyralyev and P. E. Sobolevskiĭ, Well-Posedness of Parabolic Difference Equations, Birkhäuser Verlag, Basel, 1994.
5.
S. Blunck, Maximal regularity of discrete and continuous time evolution equations, Studia Math., 146 (2001), pp. 157–176.
6.
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed., Springer, New York, 2008.
7.
R. Denk, M. Hieber, and J. Prüss, \(\mathcal{R}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), 788.
8.
K. Eriksson, C. Johnson, and S. Larsson, Adaptive finite element methods for parabolic problems VI: Analytic semigroups, SIAM J. Numer. Anal., 35 (1998), pp. 1315–1325, https://doi.org/10.1137/S0036142996310216.
9.
K. Eriksson, C. Johnson, and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér., 19 (1985), pp. 611–643.
10.
M. Geissert, Discrete maximal \(L_p\) regularity for finite element operators, SIAM J. Numer. Anal., 44 (2006), pp. 677–698, https://doi.org/10.1137/040616553.
11.
M. Geissert, M. Hess, M. Hieber, C. Schwarz, and K. Stavrakidis, Maximal \(L^p\)-\(L^q\)-estimates for the Stokes equation: A short proof of Solonnikov’s theorem, J. Math. Fluid Mech., 12 (2010), pp. 47–60.
12.
T. Kashiwabara and T. Kemmochi, Stability, analyticity, and maximal regularity for parabolic finite element problems on smooth domains, Math. Comp., 89 (2020), pp. 1647–1679.
13.
T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems, Studia Math., 234 (2016), pp. 241–263.
14.
T. Kemmochi and N. Saito, Discrete maximal regularity and the finite element method for parabolic equations, Numer. Math., 138 (2018), pp. 905–937.
15.
B. Kovács, B. Li, and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal., 54 (2016), pp. 3600–3624, https://doi.org/10.1137/15M1040918.
16.
P. C. Kunstmann, B. Li, and C. Lubich, Runge-Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity, Found. Comput. Math., 18 (2018), pp. 1109–1130.
17.
D. Leykekhman and B. Vexler, Pointwise best approximation results for Galerkin finite element solutions of parabolic problems, SIAM J. Numer. Anal., 54 (2016), pp. 1365–1384, https://doi.org/10.1137/15M103412X.
18.
D. Leykekhman and B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods, Numer. Math., 135 (2017), pp. 923–952.
19.
B. Li, Maximum-norm stability and maximal \(L^p\) regularity of FEMs for parabolic equations with Lipschitz continuous coefficients, Numer. Math., 131 (2015), pp. 489–516.
20.
B. Li, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra, Math. Comp., 88 (2019), pp. 1–44.
21.
B. Li, Maximal regularity of multistep fully discrete finite element methods for parabolic equations, IMA J. Numer. Anal., 42 (2022), pp. 1700–1734.
22.
B. Li and W. Sun, Maximal \(L^p\) error analysis of FEMs for nonlinear parabolic equations with nonsmooth coefficients, Int. J. Numer. Anal. Model., 14 (2017), pp. 670–687.
23.
B. Li and W. Sun, Maximal \(L^p\) analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra, Math. Comp., 86 (2017), pp. 1071–1102.
24.
B. Li and W. Sun, Maximal regularity of fully discrete finite element solutions of parabolic equations, SIAM J. Numer. Anal., 55 (2017), pp. 521–542, https://doi.org/10.1137/16M1071912.
25.
A. Lunardi, Interpolation Theory, 3rd ed., Edizioni della Normale, Pisa, 2018.
26.
C. Makridakis and R. H. Nochetto, A posteriori error analysis for higher order dissipative methods for evolution problems, Numer. Math., 104 (2006), pp. 489–514.
27.
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), pp. 505–521, https://doi.org/10.1137/0733027.
28.
V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Sov. Math., 8 (1977), pp. 467–529.
29.
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer-Verlag, Berlin, 2006.
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© 2024 Society for Industrial and Applied Mathematics.
History
Submitted: 20 June 2023
Accepted: 11 April 2024
Published online: 22 July 2024
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Japan Society for the Promotion of Science (JSPS): 19K14590, 21H00990
Funding: The first author was supported by JSPS KAKENHI grant 20K14357. The second author was supported by JSPS KAKENHI grants 19K14590 and 21H00990.
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