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. 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)
  2. Barrenechea, Gabriel R.; John, Volker; Knobloch, Petr: Analysis of algebraic flux correction schemes (2016)
  3. Ahmed, Naveed; Matthies, Gunar: Higher order continuous Galerkin-Petrov time stepping schemes for transient convection-diffusion-reaction equations (2015)
  4. Birken, Philipp: Termination criteria for inexact fixed-point schemes. (2015)
  5. Rang, Joachim: Improved traditional Rosenbrock-Wanner methods for stiff ODEs and DAEs (2015)
  6. Jenkins, Eleanor W.; John, Volker; Linke, Alexander; Rebholz, Leo G.: On the parameter choice in grad-div stabilization for the Stokes equations (2014)
  7. Rang, Joachim: An analysis of the Prothero-Robinson example for constructing new DIRK and ROW methods (2014)
  8. Ahmed, Naveed; Matthies, Gunar; Tobiska, Lutz: Stabilized finite element discretization applied to an operator-splitting method of population balance equations (2013)
  9. John, Volker; Novo, Julia: A robust SUPG norm a posteriori error estimator for stationary convection-diffusion equations (2013)
  10. John, Volker; Novo, Julia: On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations (2012)
  11. Matthies, Gunar; Tobiska, Lutz: A two-level local projection stabilisation on uniformly refined triangular meshes (2012)
  12. Ahmed, Naveed; Matthies, Gunar; Tobiska, Lutz: Finite element methods of an operator splitting applied to population balance equations (2011)
  13. Ahmed, N.; Matthies, G.; Tobiska, L.; Xie, H.: Discontinuous Galerkin time stepping with local projection stabilization for transient convection-diffusion-reaction problems (2011)
  14. Augustin, Matthias; Caiazzo, Alfonso; Fiebach, André; Fuhrmann, Jürgen; John, Volker; Linke, Alexander; Umla, Rudolf: An assessment of discretizations for convection-dominated convection-diffusion equations (2011)
  15. Franz, Sebastian; Matthies, Gunar: Convergence on layer-adapted meshes and anisotropic interpolation error estimates of non-standard higher order finite elements (2011)
  16. John, Volker; Knobloch, Petr; Savescu, Simona B.: A posteriori optimization of parameters in stabilized methods for convection-diffusion problems.I (2011)
  17. Franz, Sebastian; Matthies, Gunar: Local projection stabilisation on $S$-type meshes for convection-diffusion problems with characteristic layers (2010)
  18. John, Volker; Kindl, Adela: Numerical studies of finite element variational multiscale methods for turbulent flow simulations (2010)
  19. John, Volker; Kindl, Adela: A variational multiscale method for turbulent flow simulation with adaptive large scale space (2010)
  20. John, Volker; Kindl, Adela; Suciu, Carina: Finite element LES and VMS methods on tetrahedral meshes (2010)

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