Adaptive Finite Element Methods (MAGIC098)
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The main prerequisite is a strong motivation to undertake research related in modern aspects functional approximation theory, data compression, related algorithms, or the numerical analysis of partial differential equations. A solid background in undergraduate analysis and partial differential equations, some basic functional or harmonic analysis, or numerical analysis will be useful.
Autumn 2018 (Monday, October 8 to Friday, December 14)
Requirements are standard year 3 or master's level Analysis and some knowledge of elliptic partial differential equations. Exposure to Galerkin or finite element methods (as taught in spring term MAGIC-100 or equivalent) will be helpful though not essential. "Review" material will be uploaded.
We start by reviewing the standard Galerkin method with a focus on numerical approximation methods such as wavelet Galekrin, finite elements and discontinuous Galerkin for elliptic and parabolic equations, including the needed element of functional analysis, e.g., Sobolev and Besov spaces. We then recall the apriori error analysis of such methods and move onto aposteriori error analysis. We follow up this with an overview of the literature on adaptive methods and their convergence analysis with a focus on complexity of algorithms. Time allowing we look at connections between wavelet and Galerkin methods or space-time methods for parabolic (perhaps hyperbolic) problems. (NB to be reduced to 10 hours)
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Assessment will consist in a project divided in two parts. The title is Hierarchical Bank-Weiser or Verfürth estimators for P1 elements * part 1: theory (100This part consists, requires understanding and being able to work out the technical details of section 1.8 in the book of Verfürth (2013). A question that is not treated in the book will also be included. * part 2: numerics (100This part consists in a practical implementation and testing of the Heirarchical estimators with a software of your choice. Total assessment is 200
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