Uniform a priori estimates for singularly perturbed elliptic equations in multidimensions

We consider a linear singularly perturbed elliptic equation on a bounded do main in R-n. This equation serves as a model for the interplay between the transport term and a relatively small viscosity term in some problems of ma thematical physics. We establish a priori estimates that hold uniformly in the small parameter. From this one can prove an abstract a priori error est imate for the discretization with conforming finite elements. For globally directed transport fields we establish a new type of anisotropic a priori e stimate for the solution. In the case of Omega = (0, 1)(2) we prove a prior i estimates for certain higher order derivatives in isotropic as well as an isotropic norms. In a subsequent paper we will apply these results to show uniform convergence of an exponentially fitted method on rectangular grids.