A Uniform Analysis of Nonsymmetric and Coercive Linear Operators

Abstract

In this work, we show how to construct, by means of the function space interpolation theory, a natural norm $\trinorm{\cdot}$ for a generic linear coercive and nonsymmetric operator $\mathcal{L}$. The natural norm $\trinorm{\cdot}$ allows for continuity and inf-sup conditions which hold independently of $\mathcal{L}$. In particular we will consider the convection-diffusion-reaction operator, for which we obtain continuity and inf-sup conditions that are uniform with respect to the operator coefficients. In this case, our results give some insight for the analysis of the singular perturbed behavior of the operator, occurring when the diffusivity coefficient is small. Furthermore, our analysis is preliminary to applying some recent numerical methodologies (such as least-squares and adaptive wavelet methods) to this class of operators, and more generally to analyzing any numerical method within the classical framework [I. Babuska and A. K. Aziz, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972, pp. 1--359].