Abstract
This paper is devoted to obtain some norm estimates for the difference between the two resolvent operators under the discretization of the domain \(\varOmega \subset \mathbb {R}^n\), \(n\geqslant 2\), by finite element method.
Similar content being viewed by others
References
Arrieta JM, Carvalho AN, Lozada-Cruz G (2006) Dynamics in dumbbell domains. I. Continuity of the set of equilibria. J Differ Equ 231(2):551–597
Brenner SC, Scott LR (1996) The mathematical theory of finite element methods. Springer-Verlag, New York
Burenkov VI (1998) Sobolev spaces on domains. Teubner-texte zur mathematik (Teubner texts in mathematics), vol 137. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart
Carvalho AN, Cholewa JW, Lozada-Cruz G, Primo MRT (2013) Reduction of infinite dimensional systems to finite dimensions: compact convergence approach. SIAM J Math Anal 45(2):600–638
Cholewa JW, Dlotko T (2000) Global attractors in abstract parabolic problems. In: London Mathematical Society lecture note series, vol 278. Cambridge University Press, Cambridge
Figueroa-López R, Lozada-Cruz G (2014) On global attractors for a class of parabolic problems. Appl Math Inf Sci 8(2):1–8
Friedman A (2008) Partial differential equations of parabolic type. Dover Publications Inc, Mineola
Fujita H, Mizutani A (1976) On the finite element method for parabolic equations. I. Approximation of holomorphic semi-groups. J Math Soc Jpn 28(4):749–771
Fujita H, Saito N, Suzuki T (2001) Operator theory and numerical methods. In: Studies in mathematics and its applications, vol 30. North-Holland Publishing Co., Amsterdam
Gil’ MI (2012) Norm estimates for resolvents of non-selfadjoint operators having Hilbert–Schmidt inverse ones. Math Commun 17(2):599–611
Gil’ MI (2013) Resolvents of operators inverse to Schatten–von Neumann ones. Ann. Univ., Ferrara (published online December 31)
Henry D (1981) Geometric theory of semilinear parabolic equations. In: Lecture Notes in Mathematics, vol 840. Springer, New York
Pazy A (1992) Semigroups of linear operators and applications to partial differential equations. In: Applied Mathematical Sciences, 2nd edn. Springer-Verlag, New York
Zheng SM (2004) Nonlinear evolution equations. In: CRC monographs and surveys in pure and applied mathematics, vol 133. Chapman & Hall, Boca Raton
Acknowledgments
We are grateful to the anonymous referees for a number of helpful suggestions for improvement in this article. The first author was partially supported by FAPESP (Brazil) through the research grant 09/08088-9 and the second author was partially supported by FAPESP (Brazil) through the research grant 09/08435-0.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ruben Spies.
Rights and permissions
About this article
Cite this article
Figueroa-López, R., Lozada-Cruz, G. Some estimates for resolvent operators under the discretization by finite element method. Comp. Appl. Math. 34, 1105–1116 (2015). https://doi.org/10.1007/s40314-014-0168-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-014-0168-1