Nonlinear Differential Equations (NMNV406) (Summer Semester 2020/2021)
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.
You can register for the examination in SIS - currently the following dates are available:
- Friday 11.06.21 — 13:00–16:00
- Thursday 24.06.21 — 11:00–12:30 & 13:30–15:00
Further dates in July and September will be added later if necessary.
Preferably the exam will be in-person in my office (K462, 4th floor in Karlin). In order to take the exam you will need to meet the current COVID guidelines from MFF for building entry. Copy of the necessary proof will need submitting during the exam.
If you prefer to take the exam online via Zoom please email me directly.
Lectures:
- Wednesday 9:00 – 10:30, Online via Zoom (Recordings/Notes)
Due to the current Covid-19 restrictions the teaching will be online only via Zoom. Details on how to connect to the Zoom meeting will be emailled directly to all registered users (using the email address registered in SIS). If you have not received this email please contact me directly at congreve[at]karlin.mff.cuni.cz.
Practicals:
- Wednesday 10:40 – 12:10, Online via Zoom
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 ot 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