Spectral bounds of the difference Laplace operator in non-rectangular regions

We consider the eigenvalue problem for a two-dimensional difference Laplace operator in non-rectangular regions (a curvilinear triangle, a curvilinear trapezoid, a circular segment). The dependence of the eigenvalues on the parameters of the regions is elucidated. The main result is the derivation of the spectral bounds of the difference operator. A lower bound for the minimum eigenvalue and an upper bound for the maximum eigenvalue are determined. The spectral bound is determined numerically for a series of non-rectangular regions. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 94–113, 2006.