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Note that we have assumed that u(x(t), t) lies between v+ = v(x(t)+O, t) and v- = v(x(t) - 0, t) and hence, the above limit is different from zero (except, of course, if u coincides with v+ or with v- on the discontinuity curve). If we believe that what happened with the traveling wave solution also happens in the general case, this provides the explanation we were seeking since the value of the approximate solution u given by the Engquist-Osher scheme at the discontinuity of the exact solution always lies between the limits v+ and v-.

Examples of papers containing these estimates are [13], [14], [5], [19], [6], and [7] among others. The study of the sharpness of these estimates and the application of these estimates to define adaptive strategies would be very interesting and has not been done. The second problem is related to the so-called adjoint equation. In these notes, the a posteriori error estimates we have considered were obtained without solving an adjoint equation. As we have seen, the price we had to pay 44 Bernardo Cockburn was to loose control on the norm we measured the error.

I and develop an a posteriori error estimate for this type of approximate solution. To state the result, we need to introduce some notation. ))). i )) - Finally, we set Gi(T) = (0, T) X ]Rd n Gi , G(T) = U[=l Gi(T), D(T) = (0, T) X ]Rd \ G(T). With this notation, we have the following approximation result. 1. Let u be as above and let v be the entropy solution. (T)), where Residual(u) = Ut + V . 2) and U(u - c) = Iu - c I, F(u, c) = sign(u - c) (J(u) - f(c)). The interval [a, b] is the range of the initial data Vo.

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