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

Abstract

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.