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; Klöfkorn, Robert: Extendible and efficient Python framework for solving evolution equations with stabilized discontinuous Galerkin methods (2022)
  2. Dedner, Andreas; Pryer, Tristan: Discontinuous Galerkin methods for a class of nonvariational problems (2022)
  3. Dedner, Andreas; Giesselmann, Jan; Pryer, Tristan; Ryan, Jennifer K.: Residual estimates for post-processors in elliptic problems (2021)
  4. Kane, Birane: Multistage preconditioning for adaptive discretization of porous media two-phase flow (2021)
  5. Munch, Peter; Kormann, Katharina; Kronbichler, Martin: hyper.deal: an efficient, matrix-free finite-element library for high-dimensional partial differential equations (2021)
  6. Agnese, Marco; Nürnberg, Robert: Fitted front tracking methods for two-phase ncompressible Navier-Stokes flow: Eulerian and ALE finite element discretizations (2020)
  7. Church, Lewis; Djurdjevac, Ana; Elliott, Charles M.: A domain mapping approach for elliptic equations posed on random bulk and surface domains (2020)
  8. Dedner, Andreas; Klöfkorn, Robert: A Python framework for solving advection-diffusion problems (2020)
  9. Gerstenberger, Janick; Burbulla, Samuel; Kröner, Dietmar: Discontinuous Galerkin method for incompressible two-phase flows (2020)
  10. Dedner, Andreas; Kane, Birane; Klöfkorn, Robert; Nolte, Martin: Python framework for (hp)-adaptive discontinuous Galerkin methods for two-phase flow in porous media (2019)
  11. Dunbar, Oliver R. A.; Lam, Kei Fong; Stinner, Björn: Phase field modelling of surfactants in multi-phase flow (2019)
  12. Elliott, Charles M.; Fritz, Hans; Hobbs, Graham: Second order splitting for a class of fourth order equations (2019)
  13. Roberts, Nathan V.: Camellia: a rapid development framework for finite element solvers (2019)
  14. Djurdjevac, Ana; Elliott, Charles M.; Kornhuber, Ralf; Ranner, Thomas: Evolving surface finite element methods for random advection-diffusion equations (2018)
  15. Walker, Shawn W.: FELICITY: a Matlab/C++ toolbox for developing finite element methods and simulation modeling (2018)
  16. Klöfkorn, Robert; Kvashchuk, Anna; Nolte, Martin: Comparison of linear reconstructions for second-order finite volume schemes on polyhedral grids (2017)
  17. 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)
  18. 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)
  19. Bauman, Paul T.; Stogner, Roy H.: GRINS: a multiphysics framework based on the libMesh finite element library (2016) ioport
  20. Dedner, Andreas; Madhavan, Pravin: Adaptive discontinuous Galerkin methods on surfaces (2016)

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