On an optimal finite element scheme for the advection equation

In this report it is presented a numerical finite element scheme for the advection equation that attains the optimal L2L2 convergence rate O(hk+1)O(hk+1) when order kk finite elements are used, improving the order O(hk+0.5)O(hk+0.5) of other previous regularization methods. This result is also confirmed by the numerical test performed in the last section. The scheme assumes unstructured grids, periodic boundary conditions, a constant advection field and a bit (two units) stronger regularity on the exact solution than in the classical (suboptimal) finite element theory.