# Computational Methods for PDE's

## ES 105: Winter Term 2009

This is my personal web site for ES 105: Computational Methods for Partial Differential Equations with Prof. Daniel R. Lynch offered in the Winter term of 2009. On this website you will find relevant material used and presented in this course.

## Homeworks

• January 9, 2009
The file hw0.dat contains a matrix A whose size is written in the second line of the file. This is an ascii file. The following exercises are intended to get you going with fortran, matlab, and html in a unix environment. More...
• January 16, 2009

Do: 1. "Problem 12a". For the matrix solution, use the subroutine Solve.f from the 105 library. Do not change this subroutine! For the 2-D graphics, rely on the modules provided by Amir. 2) Change the periodic BC to cos(3*theta); leave all else the same. Resolve. How high is "High N"? How is this answer different than the base case?. More...

• January 23, 2009
Read about iterative methods: text, through chapter 4. Also recommended is some supplementary reading on Conjugate Direction Methods, for example in Golub and Van Loan; Trefethen and Bau; Press et al ("Recipes"); and/or Weiss.

DO: Direct Solution of Helmholtz Problem. More...

• January 30, 2009
Read: text, through Chapter 5.4 (5.5 optional).

Do: Problem 36D. This is a parabolic extension of Problem 36C (Problem Set 2). Use the same FD setup but make it time dependent and real. (For a summary of diffusion solutions, see handout.). More...

• February 6, 2009
Read: text, Part II ("The Finite Element Method"):* Chapter 7 thru 7.5* Chapter 8 - skim; read 8.7, 8.8, 8.9, 8.11
* Chapter 9 thru 9.2 Do: "Problem 42" Node and Element files are in the ENGS 105 Library, in the data section, folder hw42. Specific files are:* node file: hw5.nod ( I, x(I), y(I) )* element incidence file: hw5.ele1 (L, in(L, 1), in(L, 2), in(L, 3), in(L, 4) ). In that incidence file there is an extra incidence in(L, 4) (=1 in each case) for each element, which should be ignored; it is not needed here. . More...
• February 12, 2009: MIDTERM
The Midterm for ENGS 105 asked us to solve a Helmholtz equation using the Finite Element Method to model tidal behavior at the Bay of Fundy.
• February 20, 2009

* Text, Chapter 13
* Press Flannery etal, 2.9 and 14.3 (Singular Value Decomposition)
* LAPACK User's Guide, here (http://www.netlib.org/lapack/lug//)

Do:"Problem 44", parts (a) and (b). The mesh is here. Use mesh hw44.nod and hw44.ele. Consult hw44.README. This is an all-triangle mesh. There is a sketch with the problem; IGNORE the dimensions shown there; use those in the .nod file. The Dirichlet node numbers are in the file hw44.dnd. More...

• March 02, 2009
ES105 - Winter, 2009. Problem Set 6. Due: Monday, 2 March, 11:15 AM Read: Text, Chapter 14

Note: The recommended LAPACK routines for repetitive banded matrix solution via LU factorization are

* SGBTRF (computes LU decomposition). Call this to perform the LU factorization only.
* SGBTRS (solves assuming SGBTRF has been called). Call this every time you have a new RHS.
* Use care with the matrix storage scheme for these routines. Consult the guide here.

Note: All (x,y) units in this problem are stated in the units of the .nod file. The length scales indicated on the sketch are not to be used. Find the Sources.Given the standard solution to problem 44a. This is ``TRUTH''. Measurements of voltage are available at the following 12 locations:

* along Y=0.20: X=-0.6, -0.3, 0.0, 0.3, 0.6
* along Y=0.40: X=-0.3, 0.0, 0.3, 0.6
* along Y=0.60: X= 0.0, 0.3, 0.6

These data are available with perfect precision. There is no other data. You want to reconstruct the full solution i.e. the voltage everywhere. Neither the boundary conditions along the bottom "ground plane", nor the location or strength of the sources, are known. The Neumann BC is thought to be perfectly insulated. More...

• Final Homework Assignment: March 9, 2009
Find the Crack

(Note: All (x,y) units in this problem are stated in the units of the .nod file.)

Continuation of Homeworks 5 and 6. The node and element files are the same, Problem 44. BUT -- there is a crack in the plate! And it leaks current.

The crack is located between zones 1 and 2 of the element file; it runs from top to bottom, without interruption. It is imperfectly grounded, resulting in a leak of current out of the system. It is to be modeled as a set of point sinks of current, proportional to the local voltage.

div (t grad U) = k U delta_m

where U is the voltage, delta_m is the Dirac delta function concentrated at node m along the crack. The coefficient k is uniform along the crack. Everything else from Problem 44 is unchanged -- the ground nodes, the point source, and the boundary efflux. The current leak along the crack is new. More...

• Final Exam:
Final Exam. Earliest start date March 13, 2009. Started on March 13, 2009 at ~9 a.m. More...