Stable Discretizations of
Convection-Diffusion Problems via Computable Negative Order
Inner Products
S. Bertoluzza, C. Canuto and A. Tabacco
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 type H^(-1/2) (unlike classical SUPG methods which use elemental weighted
L^2-inner products), and the realization of this inner product via
explicitely computable multilevel decompositions of function spaces (such as
those given by wavelets or hierarchical finite elements). In the present
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.
SIAM Jour. Numer. Anal. v. 38 (2000), 1034--1055