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.

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.

DOI : 10.1051/m2an/2018040
Classification : 35K58, 65M15, 65M60
Mots clés : Parabolic semilinear equations, finite elements, Galerkin time discretization, error estimates
Meidner, Dominik 1 ; Vexler, Boris 1

1
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     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},
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     mrnumber = {3905187},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2018040/}
}
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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] G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: M2AN 38 (2004) 261–289. | DOI | Numdam | MR | Zbl

[2] A. Ashyralyev and P.E. Sobolevskiĭ 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] E. Casas, 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] E. Casas, F. Kruse and K. Kunisch, Optimal control of semilinear parabolic equations by BV-functions. SIAM J. Control Optim. 55 (2017) 1752–1788. | DOI | MR | Zbl

[5] E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21 (2002) 67–100. | MR | Zbl

[6] E. Casas, M. Mateos and A. Rösch, Finite element approximation of sparse parabolic control problems. Math. Control Rel. Fields 7 (2017) 393–417. | DOI | MR | Zbl

[7] K. Chrysafinos and L.S. Hou, 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] K. Disser, A.F.M. Ter Elst and J. Rehberg, 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] J. Elschner, J. Rehberg and G. Schmidt, Optimal regularity for elliptic transmission problems including C1 interfaces. Interfaces Free Bound. 9 (2007) 233–252. | DOI | MR | Zbl

[10] D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems. RAIRO Modél. Math. Anal. Numer. 27 (1993) 35–54. | DOI | Numdam | MR | Zbl

[11] L.C. Evans, Partial differential equations. In: Graduate Studies in Mathematics. American Mathematical Society, Providence, RI 19 (2010). | MR | Zbl

[12] B. Kovács, B. Li and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity. SIAM J. Numer. Anal. 54 (2016) 3600–3624. | DOI | MR | Zbl

[13] A. Lasis and E. Süli, hp-version discontinuous Galerkin finite element method for semilinear parabolic problems. SIAM J. Numer. Anal. 45 (2007) 1544–1569. | DOI | MR | Zbl

[14] D. Leykekhman and B. Vexler, Finite element pointwise results on convex polyhedral domains. SIAM J. Numer. Anal. 54 (2016) 561–587. | DOI | MR | Zbl

[15] D. Leykekhman and B. Vexler, Pointwise best approximation results for Galerkin finite element solutions of parabolic problems. SIAM J. Numer. Anal. 54 (2016) 1365–1384. | DOI | MR | Zbl

[16] D. Leykekhman and B. Vexler, 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] D. Leykekhman and B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods. Numer. Math. 135 (2017) 923–952. | DOI | MR | Zbl

[18] D. Leykekhman and B. Vexler, 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] D. Meidner, R. Rannacher and B. Vexler, 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] D. Meidner and B. Vexler, 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] I. Neitzel and B. Vexler, 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] J.P. Raymond and H. Zidani, Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39 (1999) 143–177. | DOI | MR | Zbl

[23] A.H. Schatz, 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] A.H. Schatz and L.B. Wahlbin, Interior maximum-norm estimates for finite element methods. II. Math. Comp. 64 (1995) 907–928. | MR | Zbl

[25] V. Thomée, 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] V. Thomée, 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] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. Johann Ambrosius Barth, Heidelberg, 2nd ed. (1995). | MR | Zbl

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