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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

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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[2]

J. Appell and P. P. Zabrejko, "Nonlinear Superposition Operators,", \textbf{95} of Cambridge Tracts in Mathematics Cambridge University Press, 95 (1990).   Google Scholar

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J.-P. Aubin and H. Frankowska, "Set-Valued Analysis,", Birkhäuser, (1990).   Google Scholar

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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

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D. Braess, "Finite Elements,", Cambridge University Press, (1997).   Google Scholar

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S. Carl and D. Motreanu, "Nonsmooth Variational Problems and their Inequalities,", Springer, (2007).   Google Scholar

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T. Donchev, Properties of one sided Lipschitz multivalued maps,, Nonlinear Analysis, 49 (2002), 13.   Google Scholar

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L. C. Evans, "Partial Differential Equations,", \textbf{19} of Graduate Studies in Mathematics. American Mathematical Society, 19 (1998).   Google Scholar

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L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis,", \textbf{9} of Series in Mathematical Analysis and Applications. Chapman & Hall/CRC, 9 (2006).   Google Scholar

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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

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S. Larsson and V. Thomée, "Partial Differential Equations with Numerical Methods,", Springer-Verlag, (2003).   Google Scholar

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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

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M. Väth, Continuity, compactness, and degree theory for operators in systems involving $p$-Laplacians and inclusions,, J. Differential Equations, 245 (2008), 1137.   Google Scholar

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R. Vinter, "Optimal Control,", Springer, (2000).   Google Scholar

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show all references

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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