Asymptotic Exactness of the Least-Squares Finite Element Residual

The discrete minimal least-squares functional $LS(f;U)$ is equivalent to the squared error $|| u-U ||^2$ 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 $LS(f;U)/||u-U||^2$ 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 $C(\mathcal{T}) LS(f;U)$ with a reliability constant $C(\mathcal{T}) \leq 1/\alpha$ 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.

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