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.

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  1. Dedner, Andreas; Giesselmann, Jan; Pryer, Tristan; Ryan, Jennifer K.: Residual estimates for post-processors in elliptic problems (2021)
  2. Kane, Birane: Multistage preconditioning for adaptive discretization of porous media two-phase flow (2021)
  3. Church, Lewis; Djurdjevac, Ana; Elliott, Charles M.: A domain mapping approach for elliptic equations posed on random bulk and surface domains (2020)
  4. Dedner, Andreas; Klöfkorn, Robert: A Python framework for solving advection-diffusion problems (2020)
  5. Gerstenberger, Janick; Burbulla, Samuel; Kröner, Dietmar: Discontinuous Galerkin method for incompressible two-phase flows (2020)
  6. Elliott, Charles M.; Fritz, Hans; Hobbs, Graham: Second order splitting for a class of fourth order equations (2019)
  7. Roberts, Nathan V.: Camellia: a rapid development framework for finite element solvers (2019)
  8. Djurdjevac, Ana; Elliott, Charles M.; Kornhuber, Ralf; Ranner, Thomas: Evolving surface finite element methods for random advection-diffusion equations (2018)
  9. Walker, Shawn W.: FELICITY: a Matlab/C++ toolbox for developing finite element methods and simulation modeling (2018)
  10. Sabir, Muhammad; Shah, Abdullah; Muhammad, Wazir; Ali, Ijaz; Bastian, Peter: A mathematical model of tumor hypoxia targeting in cancer treatment and its numerical simulation (2017)
  11. Wilbrandt, Ulrich; Bartsch, Clemens; Ahmed, Naveed; Alia, Najib; Anker, Felix; Blank, Laura; Caiazzo, Alfonso; Ganesan, Sashikumaar; Giere, Swetlana; Matthies, Gunar; Meesala, Raviteja; Shamim, Abdus; Venkatesan, Jagannath; John, Volker: ParMooN -- a modernized program package based on mapped finite elements (2017)
  12. Mitchell, Lawrence; Müller, Eike Hermann: High level implementation of geometric multigrid solvers for finite element problems: applications in atmospheric modelling (2016)
  13. Quarteroni, Alfio; Manzoni, Andrea; Negri, Federico: Reduced basis methods for partial differential equations. An introduction (2016)
  14. Kaulmann, S.; Flemisch, B.; Haasdonk, B.; Lie, K.-a.; Ohlberger, M.: The localized reduced basis multiscale method for two-phase flows in porous media (2015)
  15. Lubich, Christian; Mansour, Dhia: Variational discretization of wave equations on evolving surfaces (2015)
  16. Ohlberger, M.; Schindler, F.: Error control for the localized reduced basis multiscale method with adaptive on-line enrichment (2015)
  17. Witkowski, T.; Ling, S.; Praetorius, S.; Voigt, A.: Software concepts and numerical algorithms for a scalable adaptive parallel finite element method (2015)
  18. Brett, Charles; Elliott, Charles M.; Dedner, Andreas S.: Phase field methods for binary recovery (2014)
  19. Giesselmann, Jan; Müller, Thomas: Geometric error of finite volume schemes for conservation laws on evolving surfaces (2014)
  20. Henning, Patrick; Ohlberger, Mario; Schweizer, Ben: An adaptive multiscale finite element method (2014)

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Further publications can be found at: https://www.dune-project.org/about/publications/