ASYMPTOTIC EXACTNESS OF THE LEAST-SQUARES FINITE ELEMENT RESIDUAL

The discrete minimal least-squares functional LS(f;U) is equivalent to the squared error vertical bar vertical bar u - U vertical bar vertical bar(2) in least-squares finite element methods and so leads to an embedded reliable and efficient a posteriori error control. This paper enfolds a spectral analysis to prove that this natural error estimator is asymptotically exact in the sense that the ratio LS(f;U)/vertical bar vertical bar u - U vertical bar vertical bar(2) tends to one as the underlying mesh-size tends to zero for the Poisson model problem, the Helmholtz equation, the linear elasticity, and the time-harmonic Maxwell equations with all kinds of conforming discretizations. Some knowledge about the continuous and the discrete eigenspectrum allows for the computation of a guaranteed error bound C(T)LS(f;U) with a reliability constant C(T) <= 1/alpha smaller than that from the coercivity constant alpha. Numerical examples confirm the estimates and illustrate the performance of the novel guaranteed error bounds with improved efficiency.