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# MATH 635 Foundations of the Finite Element Method

#### Basic Information

• Manil Suri, Math/Psych 419, x52311, suri@umbc.edu,
office hours: MW 3:30-4:30 or by appointment
• Lectures: MW 5:30-6:45
• Required Text: Finite Elements by Dietrich Braess (3rd edition, previous eds OK)
• Reference Text: Finite Element Analysis by Barna Szabo and Ivo Babuska (We will refer to this for ‘p’ and ‘hp’ methods)
• Reference Text: Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson (Dover Edition) (We will use Chapters 8 and 9 for parabolic and hyperbolic problems)
• Prerequisites: A good background in mathematical and numerical analysis will be advantageous. Programming will be done in Matlab (or language of your choice).

#### Overview

Finite element methods are used to approximate the solutions of partial differential equations which arise in various engineering and other applications. This course will concentrate on the theoretical foundations of the method. The first several lectures will be devoted to developing the mathematical analysis required to analyze these methods. This will be followed by the description, error analysis and some illustrative examples with traditional ‘h’ type methods (Chapter II of the text). Following this will be a similar treatment of ‘p’ and ‘hp’ type methods (from book by Szabo and Babuska). Special attention will be paid to the approximation of singularities (such as those that develop at cracks). We will then cover topics from Chapter III of the text (nonconforming and mixed methods, Stokes problem, a posteriori error estimators) and then use Chapters 8 and 9 of the book by Johnson to cover parabolic problems and hyperbolic problems respectively. Elasticity problems will be covered, time permitting.

There will be one project, which will be to write a finite element code in one dimension, and use it to investigate some of the theory developed.

#### Learning Goals

This course has the following learning goals.

• Knowledge of the mathematical theory of variational procedures and the finite element method. The idea is to have an understanding behind the “black box” of FEM codes. (Text readings and homework problems will assist in realizing this goal.)
• Theoretical and practical understanding of convergence rates. (Text, homework and project work will assist in realizing this goal.)
• Familiarity with some of the frontiers of finite element theory and applications. (These advanced topics will be chosen according to the interests of the class.)