Stable Discretizations of Convection-Diffusion Problems via Computable Negative-Order Inner Products

A new functional framework for consistently stabilizing discrete approximations to convection-diffusion problems was recently proposed by the authors. The key ideas are the evaluation of the residual in an inner product of the type H-1/2 (unlike classical SUPG methods, which use elemental weighted L2-inner products) and the realization of this inner product via explicitly computable multilevel decompositions of function spaces (such as those given by wavelets or hierarchical finite elements). In this paper, we first provide further motivations for our approach. Next, we carry on a detailed analysis of the method, which covers all regimes (convection-dominated and diffusion-dominated). A consistent part of the analysis justifies the use of easily computable truncated forms of the stabilizing inner product. Numerical results, in close agreement with the theory, are given at the end of the paper.