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
\(U_0=U_N=0\).

\partitem{a} For the case \(\beta\) is constant, 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
this one-dimensional problem? If so, explain why this is a good idea? If not,
how would you solve it in practice?

\partitem{b} For the case that \(\beta\) is a function of \(x\) which satisfies
\(\beta(x)\geq\beta_0>0\), prove that the Jacobi iteration method converges.