Let be a Hilbert space and its dual; let , be sequences of closed subspaces of , respectively, a symmetric positive definite linear operator, and . To the equation is attached the variational form , for all , respectively for all , where is the restriction of to , namely , for all .
Suppose that the linear operators are given. If is an approximation of the solution of the equation or for all , one puts and , and defines , where denotes the projection on , namely , for all .
The paper realizes a study of the convergence rate of to u, in terms of the product of the operators defined with respect to the number of subspaces . If denotes the orthogonal projection in the subspace and then and in well specified hypotheses the above mentioned evaluations are obtained.
The applications to elliptic differential operators and a numerical model of finite element type for the plane case presented in the paper are suggestive.