SBP-FDEC: Summation-by-Parts Finite Difference Exterior Calculus

  • Gassner, Gregor (University of Cologne)
  • Bach, Daniel (University of Cologne)
  • Rueda-Ramírez, Andrés (Universidad Politécnica de Madrid)
  • Sonnendrücker, Eric (Max Planck Institute for Plasma Physics/Techn)
  • Del Rey Fernandez, David (University of Waterloo)

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In this talk, we present Summation-by-Parts (SBP) Finite Difference Exterior Calculus. The challenge with SBP-FD schemes is that typically no continuous basis function associated to the discrete integration and derivative operators are known. We will demonstrate that there is a simple matrix-relationship with the discrete SBP operators that allows us to get a discrete Vandermonde matrix to compute so-called edge degrees of freedom, used for instance in a histopolation. This simple relationship allows us to discretely construct grad, curl, div operators that are compatible analogous to the exterior calculus framework of Finite Element schemes, e.g., used to construct exactly divergence free discretizations.