A generic interface for parallel and adaptive discretization schemes: abstraction principles and the DUNE-FEM module

Starting from an abstract mathematical notion of discrete function spaces and operators, we derive a general abstraction for a large class of grid-based discretization schemes for stationary and instationary partial differential equations. Special emphasis is put on concepts for local adaptivity and parallelization with dynamic load balancing. The concepts are based on a corresponding abstract definition of a parallel and hierarchical adaptive grid given in Bastian et al. (Computing 82(2-3):103-119, 2008). Based on the abstract framework, we describe an efficient object oriented implementation of a generic interface for grid-based discretization schemes that is realized in the DUNE- FEM library (http://dune.mathematik.uni-freiburg.de). By using interface classes we manage to separate functionality from data structures. Efficiency is obtained by using modern template based generic programming techniques, including static polymorphism, the engine concept, and template metaprogramming. We present numerical results for several benchmark problems and some advanced applications.