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Math 574

Midterm # 2: due April 19

Problem 1. Consider finite difference method to solve the Laplace prob- lem

and

−uxx − uyy  = sin(πx) sin(πy) in Ω = (0, 1)2,

u = 0 on the boundary ∂Ω.

Let (xi, yj )0≤i,j≤N be uniform grids discretize the domain Ω and Let Uij de- note the approximation of u at point (xi, yj ) and Fij = f (xi, yj ).

a) Write down the finite difference method for solving the Laplace equation above.

b) In MATLAB implementation, we use the command

U  =  zeros(N+1,  N+1);

to generate an initial guess of (U 0 )0≤i,j≤N to approximate u(xi, yj ). To solve for Uij satisfies part a), write down a Gauss-Seidel iteration to generate a

(k)

ij

c) In MATLAB code, we use the command

f = @(x,y) sin(pi*x).*sin(pi*y);

to define the function f (x, y) = sin(πx) sin(πy) and use the command

to generate Fij = f (xi, yj ). Implement the Gauss Seidel iteration in part b). Set the tolerance tol = 1e-9, and

err = max

(k) (k−1)

ij ij

1≤i,j≤N −1

stop the iteration if err < tol. To plot the approximation U , use the com- mand

surf(X,Y,U)

d) For N = 8, 16, 32, record the number of the GS iteration in part c) and the error

eh  =  max

0≤i,j≤N

|u(xi, yj ) − Uij |

Note that the exact solution  u(x, y) =  1   sin(πx) sin(πy).

e) Now suppose that the error eh  ≈ Chα  for some constants C and α. Then

eh1 e

hα

≈ hα

⇒ αk =

ln eh1

ln h

− ln eh2

−

ln h

How do the values of α computed in part (d) compare to the rates of con- vergence for these norms predicted by the theory? Make sure to hand in a copy of your program as m file and the values you computed of e1/8, e1/16, e1/32  and α.

Problem  2.  a) Write a computer program to approximate the solution

of the boundary value problem:

−u   + u = 1, 0 < x < 1, u(0) = 0 u(1)  = 1.

by the finite element method using continuous piecewise linear elements on a uniform mesh of width h.  Use h = 1/100 and 1/200.

b) Let uI (x) denote the piecewise linear interpolant of u, i.e., the piecewise linear function satisfying

uI (xi) = u(xi), i = 0, · · · , N, and  e(x) = uI (x) − uh(x).

Let

E(1)  =  max

|e(xi)|,

h

E(2) =

0≤i≤N

0≤‘i≤N

/ ‘

h|e(xi)|

1/2

,

1/2

Have  the  computer  determine  E(1),  E(2),  E(3),  for  h =  1/100  and  1/200.

h h h

Note: the true solution of the boundary value problem is given by

ex e −x

u(x) = − 1 + e − 1 + e e + 1

c)

Now suppose that the error E(k)  ≈ Ckhαk  for some constants Ck and

αk.  The  error  Eh gives  an  approximation  to  lu − uhlL2(0,1)  and  the  error

Eh gives an approximation to l∇(u − uh)lL2(0,1).  How do the values of α2

and α3 computed in part (c) compare to the rates of convergence for these

norms predicted by the theory? Make sure to hand in a copy of your program as m file and the values you computed of E1/100, E1/200, and α.