On the evaluation of the eigenvalues of the finite differences laplacian over a hexagon

Abstract

Fast numerical methods for the evaluation of the eigenvalues of the finite-differences laplacian over a regular hexagon are devised. At first we show how this eigenvalue problem can be splitted into 3 eigenvalue problems, of lower dimension (reduced by a factor of 1/6), for the discrete laplacian over a regular triangle with suitable boundary conditions. Then, expressing explicitely the eigenvalues and the eigenvectors of the discrete laplacian over a triangle in terms of the coefficients of the discrete Fourier transform, we show how to deal efficiently with each subproblem. In particular we show that each step of the shifted inverse power method, for the approximation of the eigenvalues, costs O(n2log n) arithmetic operations in a sequential model of computation, and O(log n) steps with n2 processors in a parallel model of computation, where n is the number of the nodes on the edge of the hexagon. Similar estimates hold for the orthogonal iterations (subspaces iterations) method and for Lanczos method. This approach includes the deflation of the eigenvalues of the triangle from those of the hexagon. These results improve the methods given by Bauer and Reiss [1] allowing a higher precision in the approximation of the eigenvalues of the laplacian and reducing the computational cost, either in a sequential or in a parallel model of computation.