In the L-shaped region
we consider the eigenvalue problem
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