A time-adaptive space-time finite element method for incompressible Lagrangian flows with free surfaces: computational issues

Abstract

In order to further enhance the performance of the space-time Galerkin/least-squares method for solving incompressible Navier–Stokes problems involving free surfaces, issues related to the solution strategy and time adaptivity are addressed . Due to the a priori unknown boundary positions, a nonlinear system of equations normally arises at each space-time slab, which is solved by the Newton–Raphson approach. In addition, a linear system of equations for velocity and pressure that provides an alternative approach to the solution of the problem is also derived in this paper. This linear solution scheme can significantly reduce the computational costs in terms of computer CPU time and memory requirements without the sacrifice of the solution accuracy if the time-step size is sufficiently small. Furthermore, the possibility of adaptively adjusting time-step size is fully exploited. By choosing the volume loss rate as an error indicator, a simple adaptive time-stepping scheme is presented. Finally several numerical examples are provided to assess the performances of the proposed schemes.