## Homework and Projects

All homework should be emailed to me. I require that you send me your files as a single zip or gzipped tar file using the naming convention, lastname.zip or lastname.tgz unless it is a single file, which should lastname.extension, where extension makes sense based on the type of file.

I will send you a reply when I get your email. If you do not get a reply, I probably did not get your email. You are responsible for having an email account that can communicate successfully with me. I am flexible on which of my email accounts you use up to a degree. I will send you email from GMail, however. Please send email to my GMail account since that is where mail to my university address will forward to eventually.

DescriptionCategory DueWorth hw1 homework 8/31/2010 0 hw2 homework 9/2/2010 0 hw3 homework 10/5/2010 0 hw4 homework 11/2/2010 0 hw5 homework 12/?/2010 0 ## Projects

This will be detailed later in the course.

## Hw1

Fill out the course questionaire and email it back to me as an attachment. Note that it is a .doc Word document. Please return it as such (not pdf, docx, txt, rtf, xls, or anything else).

## Hw2

Read the two papers emailed and mark them up with what you do not understand, which will probably a lot. Do not worry that you do not understand the whole paper now.

## Hw3

Work out on paper the finite difference problems from 9/30/2010's class.

## Hw4

This is a continuation of Hw3's first problem only. Do the first problem over again from Hw3 with variable mesh spacing only (case H2) and use finite elements. Do not try to do this assignment in a finite difference approach by using multiple elements simultaneously (which is a complete mistake in finite element philosophy will result in having to do the assignment all over again). Do the following steps in order (similar to what is done for the CHIP problem in Chapter 4 of my textbook):

- Choose piecewise linear basis functions.
- Define a global element.
- Define mappings in both directions between a local and the global element.
- Define all of the shape functions on the global element and their drivatives.
- Define a complete set of element stiffness matrices on the global element.
- Define a complete set of element load vectors on the global element.
- Generate a sample mesh with at least 5 elements (note the word elements rather than nodal points). Create the local/global numbering table for the nodes, too.
- Show all work for generating the global stiffness matrix.
- Show all work for generating the global load vector.
- Correct the global stiffness matrix for each of the boundary conditions in Hw3's problem 1.
- Correct the global load vector for each of the boundary conditions in Hw3's problem 1.
If you really want to understand what is going on,

additionallydo the problem with piecewise constant basis functions (this is really quick to do since the basis functions are utterly trivial to work with).## Hw5

Implement Algorithm 2.4 from chapter two of my dissertation. Do this in steps:

- Choose an elliptic boundary value problem in 1D including boundary conditions.
- Choose finite elements or differences as the discretization method.
- Implement generating a stiffness matrix and load vector (finite elements) or finite difference matrix and right hand side vector for a given mesh.
- Generate a single level problem (the coarse level per se) so that there is one unknown.
- Implement a smoother and show that it works on the single level problem in no more than one iteration (thus, it is a
direct solverwhen only one unknown exists).- Refine the mesh by diving each mesh element in two equal parts (corresponding to sigma = 2 in the algorithm).
- Implement a linear interpolation algorithm for
E._{j}- Generate a second level problem (without losing the coarse level problem).
- Implement a residual calculation algorithm for a problem on level
j.- Implement a weighted averaging (or L
_{2}projection) method for the right hand side of the next coarser level.- Finish implementing the multigrid algorithm based only on the pieces implemented above.
Do steps 1-5 by Tuesday, November 23. Be prepared to discuss your implementation in class and have questions. If necessary, we will have an evening class to clarify the homework.