This course is an introduction to the finite element method as applicable to a range of problems in physics and engineering sciences. The treatment is mathematical, but only for the purpose of clarifying the formulation. The emphasis is on coding up the formulations in a modern, open-source environment that can be expanded to other applications, subsequently.

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**Syllabus**

The syllabus is outlined below. Each item is a unit consisting of between 4 and 24 segments (video lectures). A segment may be as short as one, and as long as 25 minutes. Expect to view around 3 hours of video lectures per week. There will be one paper-and-pencil, mathematical, assignment at the end of Unit 1, a quiz at the end of each Unit and 4 coding assignments. All code is to be in C++, using the open source library deal.ii.

1. Linear, elliptic partial differential equations in one dimension. Elasticity, heat conduction and mass diffusion.

2. Approximation. The finite-dimensional weak form.

3. Linear algebra; the matrix-vector form.

4. More on boundary conditions; basis functions; numerics.

5. Analysis of the finite element method.

6. Variational principles.

7. Linear, elliptic partial dierential equations for a scalar variable in three dimensions. Heat conduction and mass diffusion at steady state.

8. Lagrange basis functions and numerical quadrature in 1 through 3 dimensions.

9. Linear, elliptic, partial dierential equations for a scalar variable in two dimensions.

10. Linear, elliptic partial differential equations for vector unknowns in three dimensions (Linearized elasticity).

11. Linear, parabolic partial differential equations for a scalar unknown in three dimensions (Unsteady heat conduction and mass diffusion).

12. Linear, hyperbolic partial dierential equations for a vector unknown in three dimensions (Linear elastodynamics).

**Resources**

You can download the deal.ii library at dealii.org. The lectures include coding tutorials where we list other resources that you can use if you are unable to install deal.ii on your own computer. You will need cmake to run deal.ii. It is available at cmake.org.

**Work expectation**

You should expect to watch about 3 hours of video lectures a week. The first assignment is mathematical, to be worked out by paper and pencil. It should take a few hours, depending on your background. However, it will not be graded. Each quiz will consist of 2 or 5 multiple choice questions. The remaining are coding assignments, on which there can be quite a spread in the work you have to put in, again depending on your background. Almost all students will find the coding assignments to be more demanding of their time than the first, paper-and-pencil assignment. The second, third and fourth coding assignments should actually take no more time than the first one; you will stand to gain from a modular code, which we try to encourage in the templates that you will use.

**Books**

There are many books on finite element methods. This class does not have a required textbook. However, we do recommend the following books for more detailed and broader treatments than can be provided in any form of class:

The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, T.J.R. Hughes, Dover Publications, 2000.

The Finite Element Method: Its Basis and Fundamentals, O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu, Butterworth-Heinemann, 2005.

A First Course in Finite Elements, J. Fish and T. Belytschko, Wiley, 2007.

- Krishna Garikipati

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