A stable Petrov-Galerkin method for convection-dominated problems

In this paper, a new Petrov-Galerkin formulation for solving convection-dominated problems is presented. The method developed achieves the quasi-optimal convergence rates when the solution is regular and provides the necessary stability to avoid spurious oscillations when strong gradients are present. Such important properties allow the use of p refinement to improve the solution in regions with discontinuities because of the stability engendered by the new Petrov-Galerkin method. In this matter, a proper evaluation of the intrinsic time scale function, appearing in the design of this method, is crucial to guarantee the required accuracy.