We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations KF(x) = y. It is assumed that the available data is $y^{\delta }$ with $\left|\right|y-y^{\delta }\left|\right|\le \delta$, K: Z → Y is a bounded linear operator and F: X → Z is a nonlinear operator where X,Y,Z are Hilbert spaces. Two cases of F are considered: where $F^{\text{'}}{\left(x₀\right)}^{-1}$ exists (F’(x₀) is the Fréchet derivative of F at an initial guess x₀) and where F is a monotone operator. The parameter...

Natural superconvergence of the least-squares finite element method is surveyed for the one- and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method.