Final Project – Math 104B

Instructor: Carlos J. García Cervera

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

and ∂D is the boundary of D:

∂D = {0, 1} × [0, 1] ∪ [0, 1] × {0, 1}. (3)

We construct a two-dimensional grid: Let h = 1/n, and set

xi = ih, i = 0, 1, . . . , n,

yj = jh, j = 0, 1, . . . , n. (4)

We discretize the equation in a similar way to what we did in lecture:

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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

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