A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions

One of the most important tools for accelerating comprehensive computations in multi dimensions (in particular in 3-D) is the local adaption of the grid. In order to minimize the computing time the local grid size should be chosen such that the error ||u − uh|| between the exact solution u and the numerical solution uh is less than a given tolerance and such that the total number of cells is as small as possible. This can be obtained if an a posteriori error estimate of the form