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

TL;DR

This paper extends a Nitsche XFEM-DG space-time method for two-phase mass transport problems to three spatial dimensions, focusing on implementation details, interface approximation, and subdivision algorithms for complex integrations.

Contribution

It introduces a new subdivision algorithm for four-dimensional prisms intersected by a piecewise planar interface, enabling practical implementation in three-dimensional space.

Findings

Successful implementation of the method in three dimensions

Effective subdivision algorithm for complex space-time domains

Numerical results demonstrating method's accuracy and efficiency

Abstract

In the recent paper [C. Lehrenfeld, A. Reusken, SIAM J. Num. Anal., 51 (2013)] a new finite element discretization method for a class of two-phase mass transport problems is presented and analyzed. The transport problem describes mass transport in a domain with an evolving interface. Across the evolving interface a jump condition has to be satisfies. The discretization in that paper is a space-time approach which combines a discontinuous Galerkin (DG) technique (in time) with an extended finite element method (XFEM). Using the Nitsche method the jump condition is enforced in a weak sense. 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.