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 and Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295

References:

[1]	R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.
[2]	J. Appell and P. P. Zabrejko, "Nonlinear Superposition Operators," 95 of Cambridge Tracts in Mathematics Cambridge University Press, 1990.
[3]	J.-P. Aubin and H. Frankowska, "Set-Valued Analysis," Birkhäuser, Boston, 1990.
[4]	W.-J. Beyn and J. Rieger, An implicit function theorem for one-sided Lipschitz mappings, Set-Valued and Variational Analysis, 19 (2011), 343-359.
[5]	W.-J. Beyn and J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions, Disc. Cont. Dyn. Sys. B, 14 (2010), 409-428. doi: 10.3934/dcdsb.2010.14.409.
[6]	D. Braess, "Finite Elements," Cambridge University Press, Cambridge, 1997.
[7]	S. Carl and D. Motreanu, "Nonsmooth Variational Problems and their Inequalities," Springer, New York, 2007.
[8]	T. Donchev, Properties of one sided Lipschitz multivalued maps, Nonlinear Analysis, 49 (2002), 13-20.
[9]	L. C. Evans, "Partial Differential Equations," 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
[10]	L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," 9 of Series in Mathematical Analysis and Applications. Chapman & Hall/CRC, Boca Raton, FL, 2006.
[11]	S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis. Vol. II," 500 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 2000. Applications.
[12]	S. Larsson and V. Thomée, "Partial Differential Equations with Numerical Methods," Springer-Verlag, Berlin, 2003.
[13]	J. Rieger, Discretizations of linear elliptic partial differential inclusions, Num. Funct. Anal. Opt., 32 (2011), 904-925.
[14]	J. Rieger, Implementing Galerkin finite element methods for semilinear elliptic differential inclusions, To appear in Comp. Meth. Appl. Math.
[15]	W. Rudin, "Functional Analysis," Mc Graw Hill, Boston, 2003.
[16]	V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Number 25 in Springer Series in Computational Mathematics. Springer, 2006.
[17]	M. Väth, Continuity, compactness, and degree theory for operators in systems involving $p$-Laplacians and inclusions, J. Differential Equations, 245 (2008), 1137-1166.
[18]	R. Vinter, "Optimal Control," Springer, New York, 2000.
[19]	E. Zeidler., "Nonlinear Functional Analysis and its Applications," volume 2B. Springer, Heidelberg, 1985.

show all references

References:

[1]	R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.
[2]	J. Appell and P. P. Zabrejko, "Nonlinear Superposition Operators," 95 of Cambridge Tracts in Mathematics Cambridge University Press, 1990.
[3]	J.-P. Aubin and H. Frankowska, "Set-Valued Analysis," Birkhäuser, Boston, 1990.
[4]	W.-J. Beyn and J. Rieger, An implicit function theorem for one-sided Lipschitz mappings, Set-Valued and Variational Analysis, 19 (2011), 343-359.
[5]	W.-J. Beyn and J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions, Disc. Cont. Dyn. Sys. B, 14 (2010), 409-428. doi: 10.3934/dcdsb.2010.14.409.
[6]	D. Braess, "Finite Elements," Cambridge University Press, Cambridge, 1997.
[7]	S. Carl and D. Motreanu, "Nonsmooth Variational Problems and their Inequalities," Springer, New York, 2007.
[8]	T. Donchev, Properties of one sided Lipschitz multivalued maps, Nonlinear Analysis, 49 (2002), 13-20.
[9]	L. C. Evans, "Partial Differential Equations," 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
[10]	L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," 9 of Series in Mathematical Analysis and Applications. Chapman & Hall/CRC, Boca Raton, FL, 2006.
[11]	S. Hu and N. S. Papageorgiou, "Handbook of Multivalued Analysis. Vol. II," 500 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 2000. Applications.
[12]	S. Larsson and V. Thomée, "Partial Differential Equations with Numerical Methods," Springer-Verlag, Berlin, 2003.
[13]	J. Rieger, Discretizations of linear elliptic partial differential inclusions, Num. Funct. Anal. Opt., 32 (2011), 904-925.
[14]	J. Rieger, Implementing Galerkin finite element methods for semilinear elliptic differential inclusions, To appear in Comp. Meth. Appl. Math.
[15]	W. Rudin, "Functional Analysis," Mc Graw Hill, Boston, 2003.
[16]	V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems," Number 25 in Springer Series in Computational Mathematics. Springer, 2006.
[17]	M. Väth, Continuity, compactness, and degree theory for operators in systems involving $p$-Laplacians and inclusions, J. Differential Equations, 245 (2008), 1137-1166.
[18]	R. Vinter, "Optimal Control," Springer, New York, 2000.
[19]	E. Zeidler., "Nonlinear Functional Analysis and its Applications," volume 2B. Springer, Heidelberg, 1985.

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