L 2-Error bounds for approximate solutions of elliptic partial differential equations with Dirichlet-boundary conditions

New a posteriori (computable) upper bounds for theL ₂-norms, both ofD(u?v) and ofu?v are proposed, whereu is the exact solution of the boundary value problem $$Au: = - D(pDu) + qu = f, x in G and u = 0,x in partial G$$ andv any approximation of it (D is here the vector of partial derivatives with respect to the components ofx). It is shown that the new error bounds are better than the classical one, which is proportional to ‖Av?f‖, in many cases. This happens, e. g., ifq has some zero point inG, as in the case of a Poisson equation.