A semi-Lagrangian-Galerkin projection scheme for convection equations

We introduce in this paper a semi-Lagrangian-Galerkin projection scheme to discretize backwards in time along the characteristics the convection terms of convection diffusion equations. The scheme consists of a transport step in which the elements of the fixed mesh are transported backwards along the characteristic curves, thus generating a new mesh composed of curved elements, followed by an approximate L(2)-projection onto the finite-element space associated with the transported mesh. The new scheme is to some extent related to the so-called Lagrange-Galerkin (or characteristic-Galerkin) methods, but it may be more efficient because the number of trajectories per element to be calculated in the new scheme is smaller than that of the conventional characteristic-Galerkin scheme. it is also proved that, for linear convection problems with the velocity sufficiently smooth, the new scheme is unconditionally stable in the L(2)-norm and its order of convergence is h(m+1)/Delta t + h(2), where m is the degree of the polynomials of the finite-element space, and the velocity is in L(infinity)(0, T; W(q+1,infinity)) with integer q >= 1.