L∞- AND L2-ERROR ESTIMATES FOR A FINITE VOLUME APPROXIMATION OF LINEAR ADVECTION

We study the convergence of the upwind finite volume scheme applied to the linear advection equation with a Lipschitz divergence-free speed in R-d. We prove an h(1/2-epsilon)-error estimate in the L-infinity(R-d x [0, T])-norm for Lipschitz initial data. The expected optimal result is an h(1/2)-error estimate. In a second part, we also prove an h(1/2)-error estimate in the L-infinity(0, T; L-2(R-d))-norm for initial data in H-1(R-d).