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Iterative methods are often used for solving discretized partial differential equations. In that context a rigorous analysis of the convergence of simple methods such as the Jacobi method can be given.
As an example, consider the boundary value problem
discretized by
The eigenfunctions of the and
operator are the same:
for
the function
is an
eigenfunction corresponding to
. The
eigenvalues of the Jacobi iteration matrix
are then
.
From this it is easy to see that the high frequency modes ( i.e.,
eigenfunction with n large) are damped quickly, whereas the
damping factor for modes with n small is close to 1. The spectral
radius of the Jacobi iteration matrix is
, and it is attained for the eigenfunction
.
The type of analysis applied to this example can be generalized to
higher dimensions and other stationary iterative methods. For both the
Jacobi and Gauss-Seidel method
(below) the spectral radius is found to be where
is the
discretization mesh width, i.e.,
where
is the
number of variables and
is the number of space dimensions.