Galerkin finite element methods for semilinear elliptic differential inclusions

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March 2013, 18(2): 295-312. doi: 10.3934/dcdsb.2013.18.295

Galerkin finite element methods for semilinear elliptic differential inclusions

Wolf-Jüergen Beyn ^1, and Janosch Rieger ^2,

1.	Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld
2.	Institut für Mathematik, Universität Frankfurt, Postfach 111932, D-60054 Frankfurt a.M., Germany

Received January 2012 Revised May 2012 Published November 2012

Abstract
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In this paper we consider Galerkin finite element discretizations of semilinear elliptic differential inclusions that satisfy a relaxed one-sided Lipschitz condition. The properties of the set-valued Nemytskii operators are discussed, and it is shown that the solution sets of both, the continuous and the discrete system, are nonempty, closed, bounded, and connected sets in $H^1$-norm. Moreover, the solution sets of the Galerkin inclusion converge with respect to the Hausdorff distance measured in $L^p$-spaces.

Keywords: uncertainty quantification, finite element methods, Elliptic partial differential inclusions, set-valued numerical analysis..

Mathematics Subject Classification: Primary: 35R70, 65L60; Secondary: 35J57 34A6.

Citation: Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295

References:

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References:

[1]	R. A. Adams, "Sobolev Spaces,", Academic Press, (1975). Google Scholar
[2]	J. Appell and P. P. Zabrejko, "Nonlinear Superposition Operators,", \textbf{95} of Cambridge Tracts in Mathematics Cambridge University Press, 95 (1990). Google Scholar
[3]	J.-P. Aubin and H. Frankowska, "Set-Valued Analysis,", Birkhäuser, (1990). Google Scholar
[4]	W.-J. Beyn and J. Rieger, An implicit function theorem for one-sided Lipschitz mappings,, Set-Valued and Variational Analysis, 19 (2011), 343. Google Scholar
[5]	W.-J. Beyn and J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions,, Disc. Cont. Dyn. Sys. B, 14 (2010), 409. doi: 10.3934/dcdsb.2010.14.409. Google Scholar
[6]	D. Braess, "Finite Elements,", Cambridge University Press, (1997). Google Scholar
[7]	S. Carl and D. Motreanu, "Nonsmooth Variational Problems and their Inequalities,", Springer, (2007). Google Scholar
[8]	T. Donchev, Properties of one sided Lipschitz multivalued maps,, Nonlinear Analysis, 49 (2002), 13. Google Scholar
[9]	L. C. Evans, "Partial Differential Equations,", \textbf{19} of Graduate Studies in Mathematics. American Mathematical Society, 19 (1998). Google Scholar
[10]	L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", \textbf{9} of Series in Mathematical Analysis and Applications. Chapman & Hall/CRC, 9 (2006). Google Scholar
[11]	S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis. Vol. II,", \textbf{500} of Mathematics and its Applications. Kluwer Academic Publishers, 500 (2000). Google Scholar
[12]	S. Larsson and V. Thomée, "Partial Differential Equations with Numerical Methods,", Springer-Verlag, (2003). Google Scholar
[13]	J. Rieger, Discretizations of linear elliptic partial differential inclusions,, Num. Funct. Anal. Opt., 32 (2011), 904. Google Scholar
[14]	J. Rieger, Implementing Galerkin finite element methods for semilinear elliptic differential inclusions,, To appear in Comp. Meth. Appl. Math., (). Google Scholar
[15]	W. Rudin, "Functional Analysis,", Mc Graw Hill, (2003). Google Scholar
[16]	V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Number 25 in Springer Series in Computational Mathematics. Springer, (2006). Google Scholar
[17]	M. Väth, Continuity, compactness, and degree theory for operators in systems involving $p$-Laplacians and inclusions,, J. Differential Equations, 245 (2008), 1137. Google Scholar
[18]	R. Vinter, "Optimal Control,", Springer, (2000). Google Scholar
[19]	E. Zeidler., "Nonlinear Functional Analysis and its Applications,", volume 2B. Springer, (1985). Google Scholar

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