The Convergence of the Residual for Projective Approximations

Let uₙ denote a projective approximation to the solution u of an operator equation $Au = f$. Some general theorems on the weak and strong convergence of the residual Auₙ - f are obtained. The convergence of the residual is studied when A is an elliptic operator and the projection is onto a space of splines. Some nonuniform error estimates are obtained for finite element methods as a corollary.