First-order system least squares and the energetic variational approach for two-phase flow

Abstract

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen-Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.

Strong Solutions, Global Regularity, and Stability of a Hydrodynamic System Modeling Vesicle and Fluid Interactions

“…The energy variational approaches (in short, EnVarA) provide unified variational frameworks in studying complex fluids with micro-structures (cf. e.g., [2,24,46]). From the energetic point of view, the system (1.4)-(1.6) exhibits the competition between the macroscopic kinetic energy and the microscopic membrane elastic energy.…”

Diffuse Interface Methods for Multiple Phase Materials: An Energetic Variational Approach

“…Here, an important issue to consider is how well the numerical energy functional approximates the energy dissipation law. We mention the work in [1] where this issue is studied for a first-order system least squares finite element discretization of the magnetohydrodynamics PDE system.…”

Combining Deflation and Nested Iteration for Computing Multiple Solutions of Nonlinear Variational Problems

“…The algorithm has four main stages. The outermost phase is an NI hierarchy that has proven highly effective in reducing computational work for systems of nonlinear PDEs discretized by appropriate finite-element methods [2,4,5,6,12,37,44]. On each mesh, the algorithm first performs (undeflated) Newton iterations on interpolated versions of solutions found on the previous, coarser mesh, termed the continuation list in Algorithm 1, to further resolve the solution features on the finer mesh.…”

“…The resulting iteration is demonstrated therein to be efficient, with roughly the same amount of computational effort required to find each additional solution, and to successfully discover numerous solutions to classical problems, such as an Allen-Cahn equation and the Navier-Stokes equations. In this paper, we adapt and expand the deflation methodology by combining it with nested iteration (NI) [4,6,12,37,44], a powerful approach to reducing computational cost in the solution of nonlinear PDEs. To demonstrate the performance of the resulting algorithm, we apply it to models in liquid crystal theory, as an example of the coupled systems that arise in multiphysics simulation.…”

“…A key advantage is that these interpolated approximations are typically very good initial guesses for Newton's method on the finer grids, so very few iterations are needed on fine levels, where computation is expensive. NI strategies have been applied successfully to a variety of applications that involve coupled physics (e.g., [4,6,12,37]).…”