Extended discontinuous Galerkin methods for two‐phase flows: the spatial discretization

This work discusses a discontinuous Galerkin (DG) discretization for two‐phase flows. The fluid interface is represented by a level set, and the DG approximation space is adapted such that jumps and kinks in pressure and velocity fields can be approximated sharply. This adaption of the original DG space, which can be performed ‘on‐the‐fly’ for arbitrary interface shapes, is referred to as extended discontinuous Galerkin. By combining this ansatz with a special quadrature technique, one can regain spectral convergence properties for low‐regularity solutions, which is demonstrated by numerical examples. This work focuses on the aspects of spatial discretization, and special emphasis is devoted on how to overcome problems related to quadrature, small cut cells, and condition number of linear systems. Temporal discretization will be discussed in future works. Copyright © 2016 John Wiley & Sons, Ltd.