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 [{it P. Bastian} et al., Computing 82, No. 2--3, 103--119 (2008; Zbl 1151.65089)]. 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 (url{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.

References in zbMATH (referenced in 11 articles )

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  1. Quarteroni, Alfio; Manzoni, Andrea; Negri, Federico: Reduced basis methods for partial differential equations. An introduction (2016)
  2. Ohlberger, M.; Schindler, F.: Error control for the localized reduced basis multiscale method with adaptive on-line enrichment (2015)
  3. Witkowski, T.; Ling, S.; Praetorius, S.; Voigt, A.: Software concepts and numerical algorithms for a scalable adaptive parallel finite element method (2015)
  4. Brett, Charles; Elliott, Charles M.; Dedner, Andreas S.: Phase field methods for binary recovery (2014)
  5. Giesselmann, Jan; Müller, Thomas: Geometric error of finite volume schemes for conservation laws on evolving surfaces (2014)
  6. Brdar, S.; Dedner, A.; Klöfkorn, R.: Compact and stable discontinuous Galerkin methods for convection-diffusion problems (2012)
  7. Prud’homme, Christophe; Chabannes, Vincent; Doyeux, Vincent; Ismail, Mourad; Samake, Abdoulaye: Feel++: a computational framework for Galerkin methods and advanced numerical methods (2012)
  8. Dedner, Andreas; Klöfkorn, Robert: A generic stabilization approach for higher order discontinuous Galerkin methods for convection dominated problems (2011)
  9. Henning, Patrick: Heterogeneous multiscale finite element methods for advection-diffusion and nonlinear elliptic multiscale problems. (2011)
  10. Bastian, Peter; Heimann, Felix; Marnach, Sven: Generic implementation of finite element methods in the distributed and unified numerics environment (DUNE) (2010)
  11. Dedner, Andreas; Klöfkorn, Robert; Nolte, Martin; Ohlberger, Mario: A generic interface for parallel and adaptive discretization schemes: Abstraction principles and the DUNE-FEM module (2010)

Further publications can be found at: http://dune.mathematik.uni-freiburg.de/publications.html