Multilevel Stabilization of Convection–Diffusion Problems by Variable-Order Inner Products

We are concerned with the task of stabilizing discrete approximations to convection–diffusion problems. We propose to consistently modify the exact variational formulation of the problem by adding a fractional order inner product, involving the residual of the equation. The inner product is expressed through a multilevel decomposition of its arguments, in terms of components along a multiscale basis. The order of the inner product locally varies from −1/2 to −1, depending on the value of a suitably-defined multiscale Péclet number. Numerical approximations obtained via the Galerkin method applied to the modified formulation are analyzed. Received January 1, 2000; revised November 2, 2000