Numerics:
In this example we
investigate how our method copes with
discontinuous coefficients. We inserted a square subdomain of side 0.5
in the center of the unit square domain. We also choose the function
to be piecewise
constant and to assume the
value 100 inside the subdomain and the value 1 outside it.
The jump in the
value of
could produce a jump in the gradient of the
eigenfunctions all along the boundary of the subdomain. So the
regularity of the eigenfunctions in each subdomain, in the sense of
Assumption 1.1,
is now
between
. Using uniform refinement, the rate of convergence for
eigenvalues should be at least
or
equivalently
, where is the number of DOFs. Instead,
using our
method we obtain greater orders of convergence for big enough value of and
, as can be seen
from
Table 3.
We measure the rate of convergence computing
the value of as before. In fact
the rate of convergence for
or is
close to the rate of convergence for smooth
problems. In this case the exact eigenvalue is
unknown, but we approximate it by computing
the eigenvalue on a very fine mesh involving about half a million of
DOFs.
In Figure 3
we depict the mesh
coming from the fourth iteration of Algorithm 1
with
. This mesh is the
result of multiple refinements using both marking strategies 1 and 2
each time. As can be seen the corners of the subdomain are much more
refined than the rest of the mesh. This is clearly the effect of the
first marking strategy, since the edge residuals have detected the
discontinuity in the gradient of the eigenfunction along the interface.
Finally in Figure 4
we depict
the eigenfunction corresponding to the smallest eigenvalue of the
problem with discontinuous coefficients. This eigenfunction is the one
used to refine the mesh in Figure 3.
Figure 3:
A
refined mesh from the adaptive method corresponding to the first
eigenvalue of the problem with discontinuous coefficients.

Figure 4:
The
eigenfunction corresponding to the first eigenvalue of the problem with
discontinuous coefficients.
