The basis of mapped finite element methods are reference elements where the components of a local finite element are defined. The local finite element on an arbitrary mesh cell will be given by a map from the reference mesh cell.\parThis paper describes some concepts of the implementation of mapped finite element methods. From the definition of mapped finite elements, only local degrees of freedom are available. These local degrees of freedom have to be assigned to the global degrees of freedom which define the finite element space. We present an algorithm which computes this assignment.\parThe second part of the paper shows examples of algorithms which are implemented with the help of mapped finite elements. In particular, we explain how the evaluation of integrals and the transfer between arbitrary finite element spaces can be implemented easily and computed efficiently.

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  1. Fu, Guosheng; Han, Daozhi: A linear second-order in time unconditionally energy stable finite element scheme for a Cahn-Hilliard phase-field model for two-phase incompressible flow of variable densities (2021)
  2. Tretola, G.; Vogiatzaki, K.; Navarro-Martinez, S.: Implementation of a probabilistic surface density volume of fluid approach for spray atomisation (2021)
  3. Gamet, Lionel; Scala, Marco; Roenby, Johan; Scheufler, Henning; Pierson, Jean-Lou: Validation of volume-of-fluid OpenFOAM\circledRisoAdvector solvers using single bubble benchmarks (2020)
  4. Giussani, Filippo; Piscaglia, F.; Saez-Mischlich, G.; Hèlie, J.: A three-phase VOF solver for the simulation of in-nozzle cavitation effects on liquid atomization (2020)
  5. Grave, Malú; Camata, José J.; Coutinho, Alvaro L. G. A.: A new convected level-set method for gas bubble dynamics (2020)
  6. Claus, Susanne; Kerfriden, Pierre: A CutFEM method for two-phase flow problems (2019)
  7. Sváček, Petr: On implementation aspects of finite element method and its application (2019)
  8. Ahmed, Naveed; John, Volker; Matthies, Gunar; Novo, Julia: A local projection stabilization/continuous Galerkin-Petrov method for incompressible flow problems (2018)
  9. Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr; Rankin, Richard: A unified analysis of algebraic flux correction schemes for convection-diffusion equations (2018)
  10. de Frutos, Javier; García-Archilla, Bosco; John, Volker; Novo, Julia: Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements (2018)
  11. Ahmed, Naveed: On the grad-div stabilization for the steady Oseen and Navier-Stokes equations (2017)
  12. Ahmed, Naveed; Becher, Simon; Matthies, Gunar: Higher-order discontinuous Galerkin time stepping and local projection stabilization techniques for the transient Stokes problem (2017)
  13. Aland, Sebastian; Hahn, Andreas; Kahle, Christian; Nürnberg, Robert: Comparative simulations of Taylor flow with surfactants based on sharp- and diffuse-interface methods (2017)
  14. Bulling, Jannis; John, Volker; Knobloch, Petr: Isogeometric analysis for flows around a cylinder (2017)
  15. Ganesan, Sashikumaar; Lingeshwaran, Shangerganesh: Galerkin finite element method for cancer invasion mathematical model (2017)
  16. Giere, Swetlana; John, Volker: Towards physically admissible reduced-order solutions for convection-diffusion problems (2017)
  17. Skrzypacz, Piotr; Wei, Dongming: Solvability of the Brinkman-Forchheimer-Darcy equation (2017)
  18. Ulrich Wilbrandt, Clemens Bartsch, Naveed Ahmed, Najib Alia, Felix Anker, Laura Blank, Alfonso Caiazzo, Sashikumaar Ganesan, Swetlana Giere, Gunar Matthies, Raviteja Meesala, Abdus Shamim, Jagannath Venkatesan, Volker John: ParMooN - a modernized program package based on mapped finite elements (2017) arXiv
  19. 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)
  20. Ahmed, Naveed; Matthies, Gunar: Numerical study of SUPG and LPS methods combined with higher order variational time discretization schemes applied to time-dependent linear convection-diffusion-reaction equations (2016)

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