Some possible projects: | |

| *Abstract variational methods: Minimization of functionals* |

| This project discusses the basic lemmas behind the calculus of variations: Under which conditions does an energy functional admit a minimizer? Source: Struwe, Variational Methods, first chapter. |

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| *Variational formulations of nonlinear Laplace equations* |

| This project relates nonlinear PDEs, like the p-Laplacian, to energy minimization problems. These are important for the modelling of non-Newtonian fluids, ice and groundwater flows. Source: Renardi-Rogers, Chapter 10.2 and 10.3. |

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| *Variational inequalities (Michael) * |

| This project discusses energies with kinks (Lipschitz, not C1) as well as energy minimization under constraints (such as that the solution needs to be greater than some given function). These arise when denoising images or when a car hits a wall and are the source of some of the most important boundary conditions encountered in the real world. Source: Lecture notes |

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| *Finite elements and drums (David, Mildred and Nikoletta)* |

| Two or three talks could discuss algorithmic issues encountered in the implementation of finite elements on a computer, or show how to use freely available programs like FEniCS or FreeFEM for your own research. |

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| *Approximation properties of piecewise polynomials (Tim)* |

| How well can you approximate a function by piecewise polynomials. In the lectures we have avoided the discussion of approximation by piecewise polynomials in 2 and 3 dimensions. Explain to us the details! Source: Braess, Chapter II.6 |

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| *A posteriori error estimates and mesh refinements (Andreia and John)* |

| In the lecture we have seen a simple a posteriori error estimate for the Laplace equation. This project studies rigorous computable upper bounds on the error of the numerical solution and how to use them for adaptive refinements of the finite element mesh. Source: Lecture notes. |

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| *Discontinuous Galerkin methods (Alan)* |

| Discontinuous Galerkin methods approximate the solution to a PDE in a space of functions with discontinuities/jumps across the edges of the mesh. The variational formulation penalizes the jumps, thereby enforcing continuity in the limit as the mesh size tends to 0. The resulting methods are easily implemented and allow efficient parallelization. |

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| *Mesh refinements on the computer* |

| This project investigates how mesh refinements of a given mesh are implemented in detail. How should a triangle be divided into smaller ones? How does this work in practice on a computer? |

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| *Fast linear algebra for FEM* |

| Discuss some standard methods used to solve the finite element equations. Apart from classical linear algebra, advanced iterative or multigrid methods could be discussed. Source: Braess, Chapters IV and V, or book by Saad. |

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| *Numerical approximation of stochastic differential equations* |

| This project should present and give an analysis of basic methods for stochastic ordinary differential equations. |

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| *Stability of time-stepping methods* |

| This project gives an overview over different time stepping methods for time dependent problems (ODEs and PDEs), with a focus on stability and convergence. Source: Quarteroni, Sacco, Saleri, Numerical Mathematics. |

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| *Stochastic finite element methods* |

| Spectral stochastic finite element methods discretise the weak form of a stochastic PDE in both the spatial and stochastic dimensions. They provide an efficient alternative to Monte Carlo simulations based on solving many independent deterministic problems. This project discusses the basic formulation and analysis. |

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| *Multiscale finite element methods* |

| Many physical problems involve vastly different scales, where microscopic details influence the macroscopic behaviour (pores in a rock, cells in a cancer, high-frequency waves, atoms in a crystal). Numerical approaches at the finest scale become prohibitively expensive. Multiscale approaches use basis functions adapted to the microscopic problem (e.g. highly oscillatory) to solve the relevant PDEs on coarse mesh grids at the macroscopic scale. This project discusses the basic formulation and analysis. |

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| *Continuum mechanics* |

| Discuss the PDEs which describe the materials around us: metals, biological cells, liquid crystals, etc. Continuum mechanics provides the abstract framework to study mathematical aspects of materials. |

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