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Most finite element, or finite volume software is built around a fixed mesh data structure. Therefore, each software package can only be used efficiently for a relatively narrow class of applications. For example, implementations supporting unstructured meshes allow the approximation of complex geometries but are in gen- eral much slower and require more memory than implementations using structured meshes. In this paper we show how a generic mesh interface can be defined such that one algorithm, e. g. a discretization scheme, works on different mesh implementa- tions. For a cell centered finite volume scheme we show that the same algorithm runs thirty times faster on a structured mesh implementation than on an unstruc- tured mesh and is only four times slower than a non-generic version for a structured mesh. The generic mesh interface is realized within the Distributed Unified Numerics Environment DUNE.