Objectives: To program a 2-D finite element method; to practice proving V-ellipticity and its consequences.

(1)

Do problems 2.1, 2.2, 2.6, and 2.7 on pp. 63-64 in the text.

(2)

Consider the two-dimensional boundary value problem:

-( uxx + uyy ) = x (1-x) + y(2-y)    in  [0,1] ×[0,1]

u(x,0) = 0,   uy (x,1) = 0,   u(0,y) = 0,   ux (1,y) = -y(1 - 1 2 y) .

(If I've done the algebra correctly, the solution is u(x,y) = x(1-x)y(1-.5y).) Write a program to compute the continuous piecewise linear finite element approximation, when the region is divided into regular triangles as pictured below:

Picture Omitted
Write down the variational formulation of this problem and show that it is V-elliptic. Does the H¹-norm of the error in your computed solution seem to be of order h?