In the L-shaped region
we consider the eigenvalue problem
where 
is the Laplacian and M is a piece-wise constant function
M=1 in 
and M=M1 in 
.
We cover 
with a mesh, uniform in both directions, with the step
h = 1/(N+1). We note, that there are N mesh points on
Total number of points in our L-shaped domain is
3N**2 + 2N.
In the program
,
where n can be from 3 up to 15. To make n 16 and larger dimensions of most
arrays have to be changed.
We use the
usual five-point approximation of the Laplacian
and a one-point approximation of the operator
of multiplication by the function M.
That gives us the generalized eigenvalue problem
we solve using a preconditioned domain decomposition Lanczos method
from
A. V. Knyazev and A. L. Skorokhodov, The preconditioned gradient-type iterative
methods in a
subspace for partial generalized symmetric eigenvalue problem, SIAM J. Numerical
Analysis, v. 31, 1226,
1994
The convergence of the method is independent of N and usually
every iteration gives one correct digit in the eigenvalue.
For the particular problem we solve the cost of one iteration is O(N ln N),
and it is required to
allocate 8N memory units for components of vectors in the method.
To get such low requirements we utilize peculiarities
of the problem, for example, we use FFT to
solve problems on subdomains.
The method can be used for general symmetric eigenproblems
as well, but then it is more expensive.
The following table shows the computed smallest positive eigenvalue
for various N (down) and M1 (across):
N M1 = -1. 0. 1. 10. 100. 7 11.0209 10.6797 9.6932 1.55257 .157258 31 10.9920 10.6468 9.6562 1.55953 .157980 255 10.9793 10.6327 9.6410 1.55946 .157981 32767 10.9783 10.6315 9.6397 1.55941 .157977Some other results of numerical tests can be found in