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Elective course suitable especially for students of the Master's programmes “Mathematical Modelling in
Physics and Technology” and “Numerical and Computational Mathematics”. The aim of this course is to
showcase some of the tools and techniques that are employed in the numerical analysis of nonlinear PDE
and variational problems. After recalling some basics of the analysis of PDE and the finite element
method, in the first section of the course we will discuss some general discretisation principles for
nonlinear PDE. Subsequently, we turn to the numerical analysis of a few archetypical no
Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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There will be an oral exam at the end of the semester. The students should obtain at least 50% of the points in the problem sheets to have the right to take part in the final exam. Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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Bartels, S., Numerical Methods for Nonlinear Partial Differential Equations, Springer Series in Computational Mathematics 47, 2015.
Supplementary material will be provided during the lecture. Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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1. A crash course on the mathematical analysis of PDE. 1.1 Sobolev spaces 1.2 Variational problems and the direct method 1.3 Gradient flows 2. A crash course on the finite element method 2.1 Finite element interpolation 2.2 The linear Poisson equation 2.3 The linear heat equation 3. Proving convergence of numerical discretisation 3.1 Weak convergence, monotone operators 3.2 Gamma convergence 3.3 Error estimates for strongly convex problems 4. Applications to nonlinear problems (to choose) 3.1 The p-Laplacian 3.2 The obstacle problem 3.3 Total variation denoising 3.4 Plate bending problems 3.5 The Allen-Cahn equation 3.6 Harmonic maps Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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Fundamentals of the theory of partial differential equations, fundamentals of the finite element method in the scope of the courses Finite Element Method 1 - NMNV405 and Partial Differential Equations 1 - NMMA405. Last update: Šmíd Dalibor, Mgr., Ph.D. (22.05.2025)
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