The Nitsche XFEM-DG space-time method and its implementation in three space dimensions

We consider a standard model for mass transport across an evolving interface. The solution has to satisfy a jump condition across an evolving interface. In the recent paper [C. Lehrenfeld, A. Reusken, SIAM J. Num. Anal., 51 (2013)] a new finite element discretization method for this mass transport problem is presented and analyzed. This method is based on a space-time approach in which a discontinuous Galerkin (DG) technique (in time) is combined with an extended finite element method (XFEM). The jump condition is satisfied in a weak sense by using the Nitsche method. While the emphasis in that paper was on the analysis and one dimensional numerical experiments the main contribution of this paper is the discussion of implementation aspects for the spatially three dimensional case. As the space-time interface is typically given only implicitly as the zero-level of a level-set function, we construct a piecewise planar approximation of the space-time interface. This discrete interface is used to divide the space-time domain into its subdomains. An important component within this decomposition is a new method for dividing four-dimensional prisms intersected by a piecewise planar space-time interface into simplices. Such a subdivision algorithm is necessary for numerical integration on the subdomains as well as on the space-time interface. These numerical integrations are needed in the implementation of the Nitsche XFEM-DG method in three space dimensions. Corresponding numerical studies are presented and discussed.