562 DECOMPOSITION AND FICTITIOUS DOMAINS METHODS FOR ELLIPTIC BOUNDARY VALUE PROBLEMS S. V. Nepomnyaschikh, May 1991 Boundary value problems for elliptic second order equations in three-dimensional domains with piecewise smooth boundaries are considered. Discretization of the problem is performed using a conventional version of the finite element method with piecewise linear basis functions. The main purpose of the paper is the construction of a preconditioning operator for the resulting system of grid equations. The method is based on two approaches: decomposition of the domain into subdomains and using a new version of the method of fictitious domains. The rate of convergence of the corresponding conjugate gradient method is independent of both the grid size and the number of subdomains.