Some a priori error estimates with respect to image norms, image, for the h-extension of the finite element method in two dimensions Original Research Article

The error with respect to lower (fractional) order norms, ‖*θ‖‖*‖θ, 0<θ<10<θ<1, for the h-extension of the finite element method in 2-D, is studied and some new improved error estimates are deduced. In particular, it is shown that in polygonal domains, where the singularities dominate the regularity of the exact solution (e.g., View the MathML sourceu∈H1+δ−ɛ(Ω),∀ɛ>0,0<δ<1), the optimal rate of convergence is recovered for θ>1−δθ>1−δ. Moreover, for θ⩽1−δθ⩽1−δ the deduced error upper bound has the same order as the classical error estimate with respect to L2L2 norm (based upon the Aubin–Nitsche method). Finally, lower bound estimates of the form View the MathML source‖eh‖θ⩾C‖eh‖12, for some values of θ and positive definite unsymmetric bilinear functionals, are deduced.