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 2019 (Monday, October 7 to Friday, December 13)
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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The assessment consists of one long question addressing a research aspect of Adaptive Finite Element Methods, including aposteriori error estimation, adaptive algorithms or approximation theory. The question is divided into 8 to 10 subtasks with a mixture of analytical and computational issues to be addressed by the student. The appropriate bibliography to be used for the assessment will be provided, for you to focus on the contents rather than searching for sources. The assessment aims at gauging the overall understanding rather than the technical specifics with a pass/fail classification. 50
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