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General


Description

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.

Semester

Autumn 2018 (Monday, October 8 to Friday, December 14)

Hours

  • Live lecture hours: 10
  • Recorded lecture hours: 0
  • Total advised study hours: 40

Timetable

  • Mon 10:05 - 10:55

Prerequisites

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.

Syllabus

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)

Lecturers


Omar Lakkis (main contact)
Email O.Lakkis@sussex.ac.uk
Phone 01273 876812
Photo of Omar Lakkis
Chandrasekhar Venkataraman
Email C.Venkataraman@sussex.ac.uk
Phone 01273 876617


Bibliography


A posteriori error estimation techniques for finite element methodsRüdiger Verfürth
iFEM and adaptive finite element computational package in Matlab (R)Long Chen
Finite elements: theory, fast solvers, and applications in elasticity theoryDietrich Braess
Convergence of adaptive finite element methodsMorin, P., R. Nochetto and K. Siebert
Adaptive finite element methods with convergence ratesBinev, Peter, Wolfgang Dahmen and Ron DeVore
Approximation and learning by greedy algorithmsBarron, Andrew R. et al.
Quasi-optimal convergence rate for an adaptive fi- nite element methodCascon, J. Manuel et al.
Axioms of AdaptivityCarstensen, Carsten et al.


Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment



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

No assignments have been set for this course.

Files


Files marked L are intended to be displayed on the main screen during lectures.

Week(s)File
MAGIC098-AFEM-2018-lecturenotes.pdfL
0classplan.pdfL


Recorded Lectures


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