PRECONDITIONED GRADIENT-TYPE ITERATIVE METHODS IN A SUBSPACE FOR PARTIAL GENERALIZED SYMMETRICAL EIGENVALUE PROBLEMS

It is shown that a modification of preconditioned gradient-type iterat ive methods for partial generalized eigenvalue problems makes it possi ble to implement them in a subspace. The authors propose such methods and estimate their convergence rate. The authors also describe iterati ve methods for finding a group of eigenvalues, propose preconditioners , suggest a practical way of computing the initial guess, and consider a model example. These methods are most effective for finding minimal eigenvalues of simple discretizations of elliptic operators with piec ewise constant coefficients in domains composed of rectangles or paral lelepipeds. The iterative process is carried out on the interfaces bet ween the subdomains. Its rate of convergence does not decrease when th e mesh gets finer, and each iteration has a quite modest cost. This pr ocess is effective and parallelizable.