Enhanced Coercivity for Pure Advection and Advection–Diffusion Problems

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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.