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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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The students should obtain at least 50% of the points in the problem sheets to have the right to take part in the final project. For the final project, the students must choose a nonlinear PDE (a suitable list will be provided), implement a discretisation to approximate the solution, write down 2-3 pages describing the problem, and do a 15 minute presentation. The problem sheets and final project will contribute each 50% to the final grade. Last update: Gazca Orozco Pablo Alexei, Ph.D. (12.02.2026)
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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. Compactness methods for elliptic and parabolic PDE 1.1 Strongly monotone and Lipschitz problems: Convergence of Galerkin discretisations to minimal regularity solutions. 1.2 Quasilinear PDE: Nonlinearities induced by a maximal monotone graph. 2. Error estimates via convex duality 2.1 (Quick) Recap of convex duality: Fenchel duality and the concept of a dual problem. 2.2 Basics of a posteriori error estimation: Deriving error identities from duality principles. 2.3 Applications: The p-Laplacian, the obstacle problem, etc. 2.4 A priori error estimates via duality: Applying duality at the discrete level using the Crouzeix–Raviart and Raviart–Thomas finite elements. 3. Nonlinear solvers 3.1 Relaxed Kačanov iterations for the p-Laplacian. 3.2 Non-smooth problems I: The augmented Lagrangian method (or ADMM) for viscoplastic flow. 3.3 Non-smooth problems II: The proximal Galerkin method for the obstacle problem. 4. Other interesting PDE (if time allows) 4.1 Total variation denoising 4.2 Plate bending problems 4.3 The Allen-Cahn equation 4.4 Harmonic maps Last update: Gazca Orozco Pablo Alexei, Ph.D. (12.02.2026)
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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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