Final Project – Math 104B

Instructor: Carlos J. García Cervera

1. (30 points) Consider the following partial differential equation for u on the square

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(a). (5 points) Construct the function f(x, y) so that u(x, y) = sin(16πx(1 1

x)y(1 1 y)) solves the equation (see Figure 1).

Figure 1: Solution to equation (1).

(b). (25 points) Solve system (7) using the Conjugate Gradient Method. Set the

tolerance to 10䛈 8

, and use n = 10, 20, 40, 80, 160, 320. Plot the timings and

the number of iterations, and determine the scaling of each in terms of n. Verify

numerically that the approximation is second order accurate.

Implementaion Tips:

(a). for a simple implementation you can define the solution u (and all arrays in the

Conjugate Gradient Method) as two-dimensional arrays, just like it is defined

in (7).

Figure 2: Indexing of the nodes in the two-dimensional grid.

(b). Do not construct the full matrix associated to system (7). Instead, there are

other things you can do, for example you can define the matrix using sparse

linear algebra, or simply define a function to evaluate the matrix/vector product

directly:

2. (30 points) A physical quantity P is known to depend on the temperature T. The

file ❉❛t❛✳❝s✈ contains the experimental results. You can read the data using

(a). (20 points) Use the Least Squares method with the Householder QR to fit the

data by a polynomial model of the form

, Pi) and the results from each model for comparison.

(b). (10 points) Do a log-log plot of the same values. Explain why the log-log plot

indicates that P(T) = aTb

for some a and b. Use the Least Squares method

and solve the Normal Equations to estimate the values of a and b, and do a

regular plot and a log-log plot with the values (Ti

to show how good your approximation is. Compute the modeling error as well.