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Abstract. In this paper we shall derive a posteriori error estimates in the L 1 -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions. This result is mainly based on some fundamental a priori error estimates published in a recent paper by C. Chainais-Hillairet. The theoretical results are confirmed by numerical experiments.

We prove convergence of a class of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. The result is applied to the discontinuous Galerkin method due to Cockburn, Hou and Shu. Subject Classification (1991): 35L60, 35L65, 35L67, 65M12, 76N15 Mathematics

Abstract. We consider conservation laws on moving hypersurfaces. In this work the velocity of the surface is prescribed. But one may think of the velocity to be given by PDEs in the bulk phase. We prove existence and uniqueness for a scalar conservation law on the moving surface. This is done via a parabolic regularization of the hyperbolic PDE. We then prove suitable estimates for the solution of the regularized PDE, that are independent of the regularization parameter. We introduce the concept of an entropy solution for a scalar conservation law on a moving hypersurface. We also present some numerical experiments. As in the Euclidean case we expect discontinuous solutions, in particular shocks. It turns out that in addition to the "Euclidean shocks" geometrically induced shocks may appear.