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. 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)
  2. Ahmed, Naveed: On the grad-div stabilization for the steady Oseen and Navier-Stokes equations (2017)
  3. Bulling, Jannis; John, Volker; Knobloch, Petr: Isogeometric analysis for flows around a cylinder (2017)
  4. Ganesan, Sashikumaar; Lingeshwaran, Shangerganesh: Galerkin finite element method for cancer invasion mathematical model (2017)
  5. Giere, Swetlana; John, Volker: Towards physically admissible reduced-order solutions for convection-diffusion problems (2017)
  6. 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
  7. 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)
  8. 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)
  9. Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr: Analysis of algebraic flux correction schemes (2016)
  10. Cifani, P.; Michalek, W. R.; Priems, G. J. M.; Kuerten, J. G. M.; van der Geld, C. W. M.; Geurts, B. J.: A comparison between the surface compression method and an interface reconstruction method for the VOF approach (2016)
  11. de Frutos, Javier; García-Archilla, Bosco; John, Volker; Novo, Julia: Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements (2016)
  12. de Frutos, Javier; John, Volker; Novo, Julia: Projection methods for incompressible flow problems with WENO finite difference schemes (2016)
  13. John, Volker: Finite element methods for incompressible flow problems (2016)
  14. Ahmed, Naveed; John, Volker: Adaptive time step control for higher order variational time discretizations applied to convection-diffusion-reaction equations (2015)
  15. Ahmed, Naveed; Matthies, Gunar: Higher order continuous Galerkin-Petrov time stepping schemes for transient convection-diffusion-reaction equations (2015)
  16. Birken, Philipp: Termination criteria for inexact fixed-point schemes. (2015)
  17. John, Volker; Novo, Julia: Analysis of the pressure stabilized Petrov-Galerkin method for the evolutionary Stokes equations avoiding time step restrictions (2015)
  18. Rang, Joachim: Improved traditional Rosenbrock-Wanner methods for stiff ODEs and DAEs (2015)
  19. Caiazzo, Alfonso; Iliescu, Traian; John, Volker; Schyschlowa, Swetlana: A numerical investigation of velocity-pressure reduced order models for incompressible flows (2014)
  20. Caiazzo, Alfonso; John, Volker; Wilbrandt, Ulrich: On classical iterative subdomain methods for the Stokes-Darcy problem (2014)

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