Math 6610 • Analysis of Numerical Methods I

{Iterative Methods for Linear Systems}

Consider the boundary value problem
\[
  -u''(x)+\beta u(x)=f(x),\qquad \text{for } 0\leq x\leq1
\]
where \(\beta>0\), and with \(u(0)=u(1)=0\), and the following discretization of it:
\[
  -U_{j-1}+(2+\beta h^2)U_j-U_{j+1}=F_j
\]
for \(j=1,2,\ldots,N-1\) where \(Nh=1\), \(F_j\equiv h^2f(jh)\), and
\(U_0=U_N=0\).

Analyze the convergence properties of the Jacobi iterative method for this
problem. In particular, express the speed of convergence as a function of the
discretization stepsize \(h\). How does the number of iterations required to
reduce the initial error by a factor \(\delta\) depend on \(h\)? In practice, would
you use this method to solve the given problem? If so, explain why this is a
good idea? If not, how would you solve it in practice?

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