A mixed formulation of the Stefan problem with surface tension

Abstract

A dual formulation and finite element method is proposed and analyzed for simulating the Stefan problem with surface tension. The method uses a mixed form of the heat equation in the solid and liquid (bulk) domains, and imposes a weak formulation of the interface motion law (on the solidliquid interface) as a constraint. The basic unknowns are the heat fluxes and temperatures in the bulk, and the velocity and temperature on the interface. The formulation, as well as its discretization, is viewed as a saddle point system. Well-posedness of the time semi-discrete and fully discrete formulations is proved in three dimensions, as well as an a priori (stability) bound and conservation law. Simulations of interface growth (in two dimensions) are presented to illustrate the method.Our paper presents a completely mixed formulation of the Stefan problem, including the bulk heat equations [8]. In other words, we formulate the problem in a saddle-point framework, where the heat equations are in mixed form, and the interface motion law appears as a constraint in the system of equations with a balancing Lagrange multiplier that represents the interface temperature. To the best of our knowledge, this is a new method for the Stefan problem with surface tension. Some highlights of our method are the following.We prove that both the time semi-discrete and fully discrete systems have a priori bounds (in time) that mimic the continuous model. This assumes the interface velocity is reasonably regular and that there are no topological changes. Moreover, we can prove that both the time semi-discrete and fully discrete systems maintain conservation of thermal energy. In [5], they only achieve this for their discrete in space, continuous in time, scheme. The interface is represented by a surface triangulation that conforms to the bulk mesh which deforms with the interface. Hence, occasional re-meshing is needed, which is done by the method in [63]. One advantage of this method is that all integrals in the finite element formulation can be computed exactly. In addition, we do not need to compute the intersection of meshes at adjacent time steps to transfer solution variables from one mesh to the next (e.g. for computing L 2 projections from one mesh to another). Our method can be modified to include anisotropic surface tension via [5], which is relevant to crystal growth. The well-posedness of the method remains unchanged, as well as the a priori bound and conservation law. Other variations of the Stefan problem (e.g. Mullins-Sekerka) can be formulated with our approach by straightforward modifications. One can even include moving contact line effects when the solid phase is attached to a rigid boundary [60, 64]. SummaryIn Section 2, we describe the governing equations. Section 3 describes the fully continuous weak formulation and derives a formal a priori bound and conservation law. Section 4 explains the timediscretization and how the interface motion is handled. A variational formulation of the time semidiscrete problem is given, its ...

Semi-discrete error estimates and implementation of a mixed method for the Stefan problem

“…The formulation (4.16) was shown to be well-posed, by verifying coercivity and inf-sup conditions [10,12], in our previous work [16] with the chosen norms (4.14), (4.15). Furthermore, we showed that the semi-discrete system (4.5), (4.6) satisfies both an a priori stability bound in time and a conservation law [16].…”

“…interface motion law appears as a constraint in the system of equations with a balancing Lagrange multiplier that represents the interface temperature. Our work recently appeared in [16], where we showed that our method satisfies an a priori energy bound for the time semi-discrete and fully discrete cases. It also satisfies a conservation law for the thermal energy.…”

“…The purpose of the current paper is to give more details on the implementation of the method in [16], as well as derive preliminary (semi-discrete) error estimates between the time semi-discrete and fully discrete solutions over one time step. We discuss the difficulties in obtaining a full error analysis at the beginning of Section 6.…”

“…The particular mathematical model we consider can be found in [6,16,32]. In this section, we present the strong form of the Stefan problem in non-dimensional form.…”

“…We give the time semi-discrete weak formulation of (2.2), (2.3) which was originally presented in our previous work [16]…”