Math 6610 • Analysis of Numerical Methods I

{Iterative Methods for Linear Systems}
Consider the boundary value problem
\[
  -u''(x)=f(x),\qquad \text{for }0\le x\le1
\]
with \(u(0)=u(1)=0\), and the following discretization of it:
\[
  -U_{j-1}+2U_j-U_{j+1}=F_j
\]
for \(j=1,2,\ldots,N-1\) where \(Nh=1\) and \(F_j=h^2f(jh)\). Imagine applying the Jacobi
iterative method to solve this system of linear equations. Analyze the convergence
properties of the Jacobi method for this problem. In particular, express the speed
of convergence as a function of the discretization stepsize \(h\). You may find it
helpful to recall that the \((N-1)\times(N-1)\) matrix \(M\) defined by
\[
  m_{ij}=
  \begin{cases}
    2, & \text{if } i=j,\\
    -1, & \text{if } \lvert i-j\rvert=1,\\
    0, & \text{otherwise}
  \end{cases}
\]
has eigenvalues
\[
  \mu_l=4\sin^2\left(\frac{\pi l}{2N}\right),\qquad l=1,2,\ldots,N-1.
\]
(NOTE: This is a model problem designed to give you a simple linear system on which
to analyze the Jacobi method; one would certainly not use an iterative method to
solve a tri-diagonal system in practice!)