# American Institute of Mathematical Sciences

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

## Galerkin finite element methods for semilinear elliptic differential inclusions

 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.
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:
 [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

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
 [1] Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548 [2] Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022 [3] Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012 [4] Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019 [5] Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial & Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57 [6] Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 [7] Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115 [8] Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461 [9] Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087 [10] Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327 [11] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [12] Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1 [13] Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246 [14] Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309 [15] Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435 [16] Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807 [17] Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287-296. doi: 10.3934/proc.2015.0287 [18] Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017 [19] Thomas Lorenz. Partial differential inclusions of transport type with state constraints. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1309-1340. doi: 10.3934/dcdsb.2019018 [20] C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

2018 Impact Factor: 1.008