We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme) and with conforming finite elements in space. The main contribution of this paper is the proof of the uniform boundedness of the discrete solution. This allows us to obtain optimal error estimates with respect to various norms.
Mots clés : Parabolic semilinear equations, finite elements, Galerkin time discretization, error estimates
@article{M2AN_2018__52_6_2307_0,
author = {Meidner, Dominik and Vexler, Boris},
title = {Optimal error estimates for fully discrete {Galerkin} approximations of semilinear parabolic equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2307--2325},
publisher = {EDP-Sciences},
volume = {52},
number = {6},
year = {2018},
doi = {10.1051/m2an/2018040},
zbl = {1412.65152},
mrnumber = {3905187},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018040/}
}
TY - JOUR AU - Meidner, Dominik AU - Vexler, Boris TI - Optimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 2307 EP - 2325 VL - 52 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018040/ DO - 10.1051/m2an/2018040 LA - en ID - M2AN_2018__52_6_2307_0 ER -
%0 Journal Article %A Meidner, Dominik %A Vexler, Boris %T Optimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 2307-2325 %V 52 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018040/ %R 10.1051/m2an/2018040 %G en %F M2AN_2018__52_6_2307_0
Meidner, Dominik; Vexler, Boris. Optimal error estimates for fully discrete Galerkin approximations of semilinear parabolic equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2307-2325. doi : 10.1051/m2an/2018040. http://www.numdam.org/articles/10.1051/m2an/2018040/
[1] and , Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: M2AN 38 (2004) 261–289. | DOI | Numdam | MR | Zbl
[2] and Well-posedness of parabolic difference equations. In Vol. 698 of Operator Theory: Advances and Applications. Translated from the Russian by A. Iacob. Birkhauser Verlag, Basel (1994). | MR | Zbl
[3] , Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35 (1997) 1297–1327. | DOI | MR | Zbl
[4] , and , Optimal control of semilinear parabolic equations by BV-functions. SIAM J. Control Optim. 55 (2017) 1752–1788. | DOI | MR | Zbl
[5] and , Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21 (2002) 67–100. | MR | Zbl
[6] , and , Finite element approximation of sparse parabolic control problems. Math. Control Rel. Fields 7 (2017) 393–417. | DOI | MR | Zbl
[7] and , Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions. SIAM J. Numer. Anal. 40 (2002) 282–306. | DOI | MR | Zbl
[8] , and , Hölder estimates for parabolic operators on domains with rough boundary, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017) 65–79. | MR | Zbl
[9] , and , Optimal regularity for elliptic transmission problems including C1 interfaces. Interfaces Free Bound. 9 (2007) 233–252. | DOI | MR | Zbl
[10] and , The discontinuous Galerkin method for semilinear parabolic problems. RAIRO Modél. Math. Anal. Numer. 27 (1993) 35–54. | DOI | Numdam | MR | Zbl
[11] , Partial differential equations. In: Graduate Studies in Mathematics. American Mathematical Society, Providence, RI 19 (2010). | MR | Zbl
[12] , and , A-stable time discretizations preserve maximal parabolic regularity. SIAM J. Numer. Anal. 54 (2016) 3600–3624. | DOI | MR | Zbl
[13] and , hp-version discontinuous Galerkin finite element method for semilinear parabolic problems. SIAM J. Numer. Anal. 45 (2007) 1544–1569. | DOI | MR | Zbl
[14] and , Finite element pointwise results on convex polyhedral domains. SIAM J. Numer. Anal. 54 (2016) 561–587. | DOI | MR | Zbl
[15] and , Pointwise best approximation results for Galerkin finite element solutions of parabolic problems. SIAM J. Numer. Anal. 54 (2016) 1365–1384. | DOI | MR | Zbl
[16] and , A priori error estimates for three dimensional parabolic optimal control problems with pointwise control. SIAM J. Control Optim. 54 (2016) 2403–2435. | DOI | MR | Zbl
[17] and , Discrete maximal parabolic regularity for Galerkin finite element methods. Numer. Math. 135 (2017) 923–952. | DOI | MR | Zbl
[18] and , Discrete maximal parabolic regularity for Galerkin finite element methods for non-autonomous parabolic problems. SIAM J. Numer. Anal. 56 (2018) 2178–2202. | DOI | MR | Zbl
[19] , and , A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time. SIAM J. Control Optim. 49 (2011) 1961–1997. | DOI | MR | Zbl
[20] and , A priori error estimates for space-time finite element discretization of parabolic optimal control problems. I. Problems without control constraints. SIAM J. Control Optim. 47 (2008) 1150–1177. | DOI | MR | Zbl
[21] and , A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120 (2012) 345–386. | DOI | MR | Zbl
[22] and , Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39 (1999) 143–177. | DOI | MR | Zbl
[23] , A weak discrete maximum principle and stability of the finite element method in L∞ on plane polygonal domains. I. Math. Comp. 34 (1980) 77–91. | MR | Zbl
[24] and , Interior maximum-norm estimates for finite element methods. II. Math. Comp. 64 (1995) 907–928. | MR | Zbl
[25] , Error estimates for finite element methods for semilinear parabolic problems with nonsmooth data. In: Equadiff 6 (Brno, 1985). Vol. 1192 of Lecture Notes Math. (1986) 339–344. | MR | Zbl
[26] , Galerkin finite element methods for parabolic problems. In: Vol. 25 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2nd edition (2006). | MR | Zbl
[27] , Interpolation Theory, Function Spaces, Differential Operators. Johann Ambrosius Barth, Heidelberg, 2nd ed. (1995). | MR | Zbl
Cité par Sources :
