Some a priori error estimates with respect to Hθ norms, 0 h-extension of the finite element method in two dimensions

Authors:
G. Tsamasphyros;S. Markolefas
Affiliations:
Department of Applied Mechanics, Faculty of Applied Mathematics and Physics, National Technical University of Athens, 9 Iroon Polytechniou, Zografou 157 73, Athens, Greece;Department of Applied Mechanics, Faculty of Applied Mathematics and Physics, National Technical University of Athens, 9 Iroon Polytechniou, Zografou 157 73, Athens, Greece
Venue:
Applied Numerical Mathematics
Year:
2005

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Abstract

The error with respect to lower (fractional) order norms, ||*||θ, 0 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., u ∈ H1+δ-ε(Ω), ∀ε 0, 0 1 - δ. Moreover, for θ ≤ 1 - δ the deduced error upper bound has the same order as the classical error estimate with respect to L2 norm (based upon the Aubin-Nitsche method). Finally, lower bound estimates of the form ||eh||θ ≥ C||eh||12, for some values of θ and positive definite unsymmetric bilinear functionals, are deduced.