Article Contents
Article Contents

# Galerkin finite element methods for semilinear elliptic differential inclusions

• 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.
Mathematics Subject Classification: Primary: 35R70, 65L60; Secondary: 35J57 34A60.

 Citation:

•  [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.