On the Vreman filter based stabilization for the advection equation

This report investigates a stabilization method for first order hyperbolic differential equations. If the usual unstabilized finite element method is used to solve numerically first order hyperbolics on unstructured meshes using general elements, it is expected a loss of accuracy by one power of the mesh size and also for non-smooth solutions the computed approximation might contain spurious oscillations.To stabilize the finite element method the authors consider adding to the first order hyperbolic system, as a stabilization term, the Vreman filtering analog of the VMS (variational multiscale method), Vreman (2003) [22].The resulting model is shown to have the expected highest attainable convergence rate, O(hk+0.5), when the filter radius δ is optimally chosen. The theoretical rates are also checked numerically and comparison with another known scheme is provided.