SPECTRAL DISTRIBUTION OF PRECONDITIONED ELLIPTIC-OPERATORS

This paper is concerned with the singular value distribution and eigenvalue distribution of preconditioned non-selfadjoint elliptic operators corresponding to finite element discretizations of elliptic boundary value problems. The distribution of the singular values has an important bearing on the convergence behavior of the conjugate gradient method applied to the normal equations. The eigenvalue distribution has an important bearing on various alternative conjugate gradient like iterative methods. The main results give a precise interval on which the singular values cluster. It is also shown that the (complex) eigenvalues cluster about an interval of the real axis under suitable assumptions. The analysis is based on the theory of collectively compact operators.