Enhanced Coercivity for Pure Advection and Advection–Diffusion Problems

We present a common framework in which to set advection problems or advection–diffusion problems in the advection dominated regime, prior to any discretization. It allows one to obtain, in an easy way via enhanced coercivity, a bound on the advection derivative of the solution in a fractional norm of order −1/2. The same bound trivially applies to any Galerkin approximate solution, yielding a stability estimate which is uniform with respect to the diffusion parameter. The proposed formulation is discussed within Fourier methods and multilevel (wavelet) methods, for both steady and unsteady problems.