Nonlinear Differential Equations (NMNV406) (Summer Semester 2021/2022)
This lecture course will cover the solution of nonlinear differential equations. Topics covered will include:
- Basic theorems from the theory of monotone and potential operators,
- Nonlinear differential equations in divergent form,
- Carathéodory's growth conditions, Nemycky operators,
- Variational methods and application of theory of monotone and potential operator, and proof of existence of solution,
- Numerical solution of nonlinear differential equations using the finite element method.
Exams:
The final exam will consist of a 30 minute oral examination on the topics covered.
Lectures:
- Tuesday 15:40 – 17:10, K5 Sokolovská 83 Karlín
Practicals:
- Tuesday 17:20 – 18:50, K5 Sokolovská 83 Karlín
Suggested Reading:
- K. Böhmer, Numerical Methods for Nonlinear Elliptic Differential Equations, Oxford University Press, 2010.
- V. Dolejší & K. Najzar, Nelineární funkcionální analýza, matfyzpress, 2011.
- E. Ziedler. Nonlinear functional analysis and its applications I, Springer, 1984.
- E. Ziedler. Nonlinear functional analysis and its applications II/A, Springer, 1990.
- J. Nečas. Introduction to the Theory of Nonlinear Elliptic Equations, Wiley, 1986
- L. C. Evans, Partial Differential Equations, AMS, 2010.
Additional journal articles for section 4:
- S. Congreve and T. P. Wihler. Iterative Galerkin discretizations for strongly monotone problems. J. Comput. Appl. Math., 311:457–472, 2017. DOI 10.1016/j.cam.2016.08.014.
- W. B. Liu and John W. Barrett. Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities. ESAIM Modél. Math. Anal. Numér., 26(6):725–744, 1994. DOI 10.1051/m2an/1994280607251.
- M. Amrein and T. P. Wihler. An adaptive Newton-method based on a dynamical systems approach. Commun. Nonlinear Sci., 19(9):2958–2973, 2014. DOI 10.1016/j.cnsns.2014.02.010
- M. Amrein and T. P. Wihler. Fully adaptive Newton-Galerkin methods for semilinear elliptic partial differential equations. SIAM J. Sci. Comput., 37(4):A1637–A1657, 2015. DOI 10.1137/140983537