Andrew Knyazev, Software. Math UC Denver

Andrew Knyazev , Software & Simulation
Eigenvalues of Laplacian in the L-shaped domain

In the L-shaped region

displaymath11

we consider the eigenvalue problem

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where tex2html_wrap_inline15 is the Laplacian and M is a piece-wise constant function M=1 in tex2html_wrap_inline17 and M=M1 in tex2html_wrap_inline19 .
We cover tex2html_wrap_inline21 with a mesh, uniform in both directions, with the step h = 1/(N+1). We note, that there are N mesh points on tex2html_wrap_inline23 Total number of points in our L-shaped domain is 3N**2 + 2N.
In the program tex2html_wrap_inline25 , 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  .157977
Some other results of numerical tests can be found in
A. L. Skorokhodov, Domain decomposition method in a partial symmetric eigenvalue problem, In: Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Eds: Glowinski, Kuznetsov, Meurant, Periaux and Widlund, SIAM, 1991, 82-87.
The method also computes the trace of the corresponding eigenvector on tex2html_wrap_inline23
The FORTRAN-77 code (written by A.L. Skorokhodov and A. Knyazev in 1990) is available here. Feel free to copy/modify/sell it as far as you keep a reference to Andrew Knyazev (knyazev@na-net.ornl.gov).


Department of Mathematics, University of Colorado Denver
P.O. Box 173364, Campus Box 170, Denver, CO 80217-3364.
Street Address: 1250 14th Str. Room 644, Denver CO 80202
Phone: (303) 556-8442. Fax: (303) 556-8550
This material is based upon work supported by the National Science Foundatio n under Grant No. DMS 9501507. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Andrew Knyazev, October 13, 1996