On Dirichlet boundary condition in spectral problems for grid elliptic operators (Q1964469)

Spectral problems for grid elliptic operators are considered. It is well known that one of the most difficult and most famous problems of mathematical physics and computional mathematics is the problem of constructing approximations from below for eigenvalues of symmetrical positive elliptical operators with Dirichlet boundary conditions (the classical approximations of Rayleigh-Ritz type are convergent to the eigenvalues from above). Some new modifications of the classical grid-projective approximations of the spectral problems are presented. It is noted that the new modifications give new approximate eigenvalues which are approximations from below for the eigenvalues of standard grid operators.