Mathematical Theory of Finite Elements

Wykładowca: Prof. Leszek Demkowicz
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W dniach 25‐29 czerwca 2018 na Wydziale Inżynierii Lądowej Politechniki Krakowskiej Profesor Leszek Demkowicz z University of Texas at Austin przeprowadzi kurs Mathematical Theory of Finite Elements (A crash course for engineers) w ramach Studiów Doktoranckich WIL PK. W imieniu organizatorów kursu i Profesora Demkowicza serdecznie zapraszamy wszystkich zainteresowanych tą tematyką członków PTMKM do nieodpłatnego uczestnictwa.

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We would like to inform that Prof. Leszek Demkowicz will give the course Mathematical Theory of Finite Elements (A crash course for engineers) in Cracow University of Technology, The Faculty of Civil Engineering on 25-29 June 2018. The course is for PhD students as well as for anyone interested. Attendance at the course is free of charge. All lectures will be given in English. The syllabus is attached. All participants are kindly requested to make their own travel and accommodation arrangements. The participants should register by e-mail to Prof. Witold Cecot: Witold.Cecot@L5.pk.edu.pl, by June 11, 2018.

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Abstract

We review fundamentals of Galerkin and conforming Finite Element (FE) methods using the model diffusion-convection-reaction problem. We discuss the possibility of different variational formulations leading to different energy spaces and corresponding conforming elements. The course is focusing on the famous inf-sup stability condition and the concept of discrete stability. We review the classical results of Babu ́ska, Mikhlin and Brezzi, and finish the exposition with fundamentals of the Discontinuous Petrov Galerkin (DPG) method. The week-long course consists of three 1.5 hour lectures per day accompanied with a one hour afternoon Q/A discussion session.

Course program

Day 1
1. Classical calculus of variations. Concept of a variational formulation.
2. Diffusion-convection-reaction model problem. Different variational formulations.
3. Distributional derivatives and different energy spaces.
Day 2
1. Abstract framework: vector space, linear and bilinear forms, dual space.
2. Galerkin and Riesz methods.
3. Exact sequence elements.
Day 3:
1. Banach Closed Range, Babuška-Nečas, and Babuška Theorems.
2. Coercivity. Lax-Milgram Theorem and Cea’s Lemma.
3. Well posedness of the variational formulations for the model problem.
Day 4:
1. Mikhlin’s theory of asymptotic stability and convergence.
2. Brezzi’s theory of mixed problems.
3. Concept of optimal test functions.
Day 5:
1. Breaking test spaces and bilinear forms.
2. Fundamentals of the Discontinuous Petrov-Galerkin (DPG) Method.
3. Current research on the DPG method.

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