A novel arbitrary Lagrangian–Eulerian finite element method for a parabolic/mixed parabolic moving interface problem

Highlights

• Arbitrary Lagrangian–Eulerian (ALE)-finite element method.

• Parabolic/mixed parabolic moving interface problem with jump coefficients.

• A novel Piola-type arbitrary Lagrangian–Eulerian mapping.

• H(div)-invariance of the novel ALE mapping along the time.

• Stability and error estimates in the ALE frame are analyzed for the mixed finite element approximation.

Abstract

In this paper, a monolithic arbitrary Lagrangian–Eulerian (ALE)-finite element method (FEM) is developed based upon a novel ALE mapping for a type of parabolic/mixed parabolic moving interface problem with jump coefficients. A stable Stokes-pair mixed FEM within a specific stabilization technique and a novel ALE time-difference scheme are developed to discretize this interface problem in both semi- and fully discrete fashion, for which the stability and error estimate analyses are conducted in the ALE frame. Numerical experiments are carried out to validate all theoretical results in different cases. The developed novel ALE-FEM can be extended to a moving interface problem that involves the pore fluid (Darcy) equation or Biot’s model in the future.

Introduction

The following parabolic/mixed parabolic moving interface problem with jump coefficients is considered in this paper:

$$\frac{\partial u_1}{\partial t}-\nabla \cdot ({\beta }_1\nabla u_1)=f_1,\text{in}{\Omega }_1^t\times (0,T],$$$$\frac{\partial u_2}{\partial t}-\nabla \cdot {\mathit{v}}_2=f_2,\text{in}{\Omega }_2^t\times (0,T],$$$${\mathit{v}}_2={\beta }_2\nabla u_2,\text{in}{\Omega }_2^t\times (0,T],$$$$u_1(\mathit{x},t)=g_1,\text{on}\partial {\Omega }_1^t\setminus {\Gamma }^t,$$$$u_2(\mathit{x},t)=g_2,\text{on}\partial {\Omega }_2^t\setminus {\Gamma }^t,$$$$u_1(\mathit{x},0)=u_1^0\left(\mathit{x}\right),\text{in}{\Omega }_1^0={\hat{\Omega }}_1,$$$$u_2(\mathit{x},0)=u_2^0\left(\mathit{x}\right),\text{in}{\Omega }_2^0={\hat{\Omega }}_2,$$$$u_1(\mathit{x},t)=u_2(\mathit{x},t),\text{on}{\Gamma }^t,$$$${\beta }_1\nabla u_1\cdot {\mathit{n}}_1+{\mathit{v}}_2\cdot {\mathit{n}}_2=0,\text{on}{\Gamma }^t,$$

which is a reformulation of the original parabolic moving interface problem with the parabolic equation on one side of the interface being rewritten as its mixed form (2), (3) in terms of the scalar-valued variable $u_2$ and the vector-valued variable ${\mathit{v}}_2$, while leaving the parabolic equation on the other side of the interface unchanged as (1) in terms of the scalar-valued variable $u_1$. Here, $\Omega$ is an open bounded domain in ${\mathbb{R}}^d(d=2,3)$ with a convex polygonal boundary $\partial \Omega$, and consists of two subdomains ${\Omega }_i^t≔{\Omega }_i\left(t\right)\subset \Omega (i=1,2)$ separated by a smooth interface ${\Gamma }^t≔\Gamma \left(t\right)\in C^2$ which may move/deform with time $t\in [0,T](T>0)$, satisfying that $\overline{\Omega }=\overline{{\Omega }_1^t}\cup \overline{{\Omega }_2^t}$, ${\Omega }_1^t\cap {\Omega }_2^t=0̸$, ${\Gamma }^t=\partial {\Omega }_1^t\cap \partial {\Omega }_2^t$. The scalar-valued solution $u$ that is defined in $\Omega \times [0,T]$ satisfies $u{|}_{{\Omega }_i^t}=u_i(i=1,2)$, and is associated with the source term $f\in L^2(0,T;L^2\left(\Omega \right))$ that satisfies $f{|}_{{\Omega }_i^t}=f_i\in L^2(0,T;L^2\left({\Omega }_i^t\right))(i=1,2)$. The jump piecewise constant $\beta$ satisfies $\beta {|}_{{\Omega }_i^t}={\beta }_i(i=1,2)$, where ${\beta }_1\ne {\beta }_2$ and $min({\beta }_1,{\beta }_2)\ge C>0$. As illustrated in Fig. 1, two subdomains ${\Omega }_1^t$ and ${\Omega }_2^t$, which follow the interface motion of ${\Gamma }^t$ and may move/deform along $t\in [0,T]$, are termed as the current (Eulerian) domains with respect to $\mathit{x}$, in contrast to their initial (Lagrangian or reference) domains, ${\hat{\Omega }}_i≔{\Omega }_i^0(i=1,2)$ with respect to $\hat{\mathit{x}}$. Non-homogeneous boundary conditions and initial conditions are given in (4)–(7), in addition, no-slip type of interface conditions are given in (8), (9) that make sure both the scalar-valued primary solution and the flux are continuous across the interface ${\Gamma }^t$.

In practice, (1)–(9) can be generalized as a type of coupled multiphysics problem whose two subproblems are defined in two subdomains and interacts with each other through a moving interface. For instance, the mixed parabolic equations (2), (3) can be extended to the pore fluid (Darcy) equation in which $u_2$ and ${\mathit{v}}_2$ play the role of pore fluid pressure and velocity, respectively. In that realistic scenario, ${\beta }_2=\frac{K}{\mu \varphi }$, standing for the ratio between the permeability $K$ and the product of the viscosity $\mu$ and the porosity $\varphi$ of the pore fluid. Similarly, the standard parabolic equation (1) can be also generalized as a diffusion–reaction equation with only respect to the pressure of another specific kind of pore fluid defined in a different type of porous medium, where the parameter ${\beta }_1$ is formed by largely different permeability, viscosity and porosity in contrast to the ones on the other side of the interface. Especially, if the Biot’s equation [1], [2], [3], [4] is involved in this coupled multiphysics problem on one side of the interface, then the pore fluid equation (2), (3) with respect to both the pressure and the velocity must be contained in the model description, and cannot be combined together to become a single diffusion–reaction (parabolic) equation with respect to the pore fluid pressure only.

In summary, the practical problems introduced at above are closely relevant with but more complicated than the present parabolic/mixed parabolic interface problem (1)–(9). However, the commonly essential part of those practical problems is basically represented by the simplified problem (1)–(9), that is, one subproblem on one side of the interface belongs to the $H\left(\text{div}\right)$-type mixed problem that involves both the vector-valued solution in $H\left(\text{div}\right)$ space and the scalar-valued solution in $L^2$ space, while the other subproblem on the other side of the interface simply belongs to the $H^1$-type problem that involves at least one solution in $H^1$ space, as shown in Section 3. Therefore, we choose to study the simplified but also typical model (1)–(9) in this paper in order to develop and analyze a novel finite element method for dealing with $H\left(\text{div}\right)$-function in a moving subdomain that interacts with $H^1$-function on the other moving subdomain through the moving interface, essentially. The research results found in this paper will inspire us to work on the Stokes/Biot moving interface problem in the future which involves the Darcy’s fluid and Newtonian fluid together and interacts through the moving interface.

To numerically solve unsteady moving domain/interface problems such as (1)–(9) in an accurate and efficient fashion, in this paper we adopt the interface-fitted mesh method that has become the most reliable numerical approach due to its naturally high accuracy, as long as the moving mesh, which adapts to the moving boundary/interface all the time, can be efficiently generated. For that purpose, the arbitrary Lagrangian–Eulerian (ALE) method [5], [6], [7], [8], [9] has been adopted as the most popular approach of body-fitted mesh methods due to its high feasibility, where the mesh on the interface is accommodated to be shared by both subproblems on either side of the interface, and thus to automatically satisfy interface conditions as sketched in (8), (9), all the time. In the past few decades, the ALE-finite element method (ALE-FEM) has been analyzed for the parabolic problem in a moving domain [10], [11] where the stability and optimal error estimates are derived, for the Stokes problem in a moving domain [12] where a suboptimal error estimate is obtained in the energy norm, and for the Stokes/parabolic interface problem [13] in which the stability and the optimal error estimate in the energy norm are derived for both semi- and full discretizations. But, to the best of our knowledge, so far the ALE-FEM has not yet been applied to a mixed parabolic equation that is involved in a moving interface problem as described in (1)–(9). Moreover, we find out the classical ALE mapping technique fails in tackling such moving interface problem that involves the mixed parabolic equations, a novel ALE mapping needs to be developed first in order to reapply the ALE-FEM to (1)–(9), as explained below.

The core of the classical ALE method is the introduction of ALE mapping, which is essentially a type of time-dependent and bijective affine mapping, ${\mathit{X}}_i^t\in {\left(W^{2,\infty }\left({\hat{\Omega }}_i\right)\right)}^d$, defined from the initial (Lagrangian) domain ${\hat{\Omega }}_i$ to the current (Eulerian) domain ${\Omega }_i^t(i=1,2)$, as follows [10], [11], [12]

$$\begin{array}{ccc}\hfill {\mathit{X}}_i^t:{\hat{\Omega }}_i& \hfill \to \hfill & {\Omega }_i^t,\hfill \\ \hfill {\hat{\mathit{x}}}_i& \hfill \mapsto \hfill & {\mathit{x}}_i({\hat{\mathit{x}}}_i,t),i=1,2,\hfill \end{array}$$

and ${\left({\mathit{X}}_i^t\right)}^{-1}\in {\left(W^{1,\infty }\left({\Omega }_i^t\right)\right)}^d$. The key reason that this ALE mapping works well for the standard parabolic moving boundary/interface problem, such as (1) in ${\Omega }_1^t\times (0,T]$, is that the property of $H^1$-invariance holds for $u_1=u_1({\mathit{x}}_1({\hat{\mathit{x}}}_1,t),t)$ and its ALE time derivative, ${\frac{\partial u_1}{\partial t}|}_{{\hat{\mathit{x}}}_1}=\frac{\partial u_1}{\partial t}({\mathit{x}}_1,t)+{\mathit{w}}_1({\mathit{x}}_1,t)\cdot \nabla u_1({\mathit{x}}_1,t)$, at any time $t\in [0,T]$, where ${\mathit{w}}_1({\mathit{x}}_1,t)=\frac{\partial {\mathit{X}}_1^t}{\partial t}\circ {\left({\mathit{X}}_1^t\right)}^{-1}$ is the domain velocity of ${\Omega }_1^t$. This important property is shown in the following Proposition 1.1.

Proposition 1.1 [10], [11]

For any $t\in (0,T]$, $u_1({\mathit{x}}_1,t)\in H^1\left({\Omega }_1^t\right)$ and ${\frac{\partial u_1}{\partial t}|}_{{\hat{\mathit{x}}}_1}\in H^1\left({\Omega }_1^t\right)$ if and only if ${\hat{u}}_1({\hat{\mathit{x}}}_1,t)=u_1({\mathit{x}}_1,t)\circ {\mathit{X}}_1^t\in H^1\left({\hat{\Omega }}_1\right)$, and vice versa.

However, such defined ALE mapping, ${\mathit{X}}_i^t$, only holds the $H^1$-invariance but not the $H\left(\text{div}\right)$-invariance in ${\Omega }_i^t(i=1,2)$, i.e., for ${\hat{\mathit{v}}}_2({\hat{\mathit{x}}}_2,t)={\mathit{v}}_2({\mathit{x}}_2,t)\circ {\mathit{X}}_2^t\in H(\text{div};{\hat{\Omega }}_2)$ that is defined in (3), Proposition 1.1 cannot guarantee ${\mathit{v}}_2({\mathit{x}}_2,t)\in H(\text{div};{\Omega }_2^t)$ for any $t\in (0,T]$. Thus, the affine-type ALE mapping, ${\mathit{X}}_i^t(i=1,2)$, cannot directly work for the parabolic/mixed parabolic interface problem (1)–(9) whose weak form involves the space $H(\text{div};{\Omega }_2^t)\times L^2\left({\Omega }_2^t\right)$ for $({\mathit{v}}_2,u_2)$. In general, for any function ${\hat{\mathit{v}}}_2\in H(\text{div};{\hat{\Omega }}_2)$, the aforementioned affine-type ALE mapping, ${\mathit{X}}_2^t$, cannot guarantee ${\mathit{v}}_2={\hat{\mathit{v}}}_2\circ {\left({\mathit{X}}_2^t\right)}^{-1}\in H(\text{div};{\Omega }_2^t)$, and vice versa. If this is not guaranteed, then, as significantly shown in Proposition 1.1 for the success of ALE method for $H^1$-type moving boundary/interface problems, the ALE technique will fail in producing a stable and convergent finite element approximation to $H\left(\text{div}\right)$-type moving boundary/interface problems, for instance, the parabolic/mixed parabolic interface problem (1)–(9). Thus, a new idea and a novel development of ALE approach within the frame of body-fitted mesh methods are urgently demanded in order to successfully apply the ALE-FEM to the present parabolic/mixed parabolic interface problem, or to the Stokes/Biot moving interface problems in the future, which are all associated with $H\left(\text{div}\right)$ space and all need the $H\left(\text{div}\right)$-invariance property to guarantee a stable and convergent ALE finite element approximation.

Our goal in this paper is to introduce a novel ALE mapping, ${\mathcal{P}}^t:{\hat{\Omega }}_2\to {\Omega }_2^t$, to the mixed parabolic part in ${\Omega }_2^t$ based upon the combination of the classical affine-type ALE mapping and a specific transformation called the Piola transformation. We will prove that this new ALE mapping ${\mathcal{P}}^t$, named as Piola-type ALE mapping, guarantees the $H\left(\text{div}\right)$-invariance in ${\Omega }_2^t$, i.e., for any ${\hat{\mathit{v}}}_2\in H(\text{div};{\hat{\Omega }}_2)$, ${\mathit{v}}_2={\mathcal{P}}^t\left({\hat{\mathit{v}}}_2\right)\in H(\text{div};{\Omega }_2^t)$ and ${\frac{\partial {\mathit{v}}_2}{\partial t}|}_{{\hat{\mathit{x}}}_2}={\mathcal{P}}^t\left(\frac{\partial {\hat{\mathit{v}}}_2}{\partial t}\right)\in H(\text{div};{\Omega }_2^t)$, and vice versa. On account of this Piola-type ALE mapping, we will develop a novel ALE-mixed finite element method for the present parabolic/mixed parabolic interface problem, then analyze its stability and convergence properties for both semi- and fully discrete schemes in a properly chosen mixed finite element space. In fact, since tackling the Stokes/Biot moving interface problem is our next goal, to cooperate with the Stokes problem on one side of the interface as well as to comply with the body-fitted mesh method, one shall discretize the mixed parabolic problem on the other side of the interface using the same mixed finite element as for the Stokes problem, i.e., a stable Stokes-pair. To that end, we need to first introduce a stabilization technique for the mixed parabolic problem in order to make the $H^1$-type Stokes-pair mixed finite element work for the $H\left(\text{div}\right)$-type mixed parabolic equations. Similar stabilization technique has been done for the mixed elliptic problem [14] and the mixed parabolic problem [15]. Basically, a term ${(\nabla \cdot {\mathit{v}}_2,\nabla \cdot {\tilde{\mathit{v}}}_2)}_{{\Omega }_2^t}$ needs to be properly introduced into the original weak form of the mixed parabolic equation in ${\Omega }_2^t$ while the consistency between them is still preserved. Then, we adopt a stable Stokes-pair, e.g., MINI mixed element, to discretize the stabilized mixed finite element scheme, considering that the regularity of solution $u$ to the moving interface problem (1)–(9) is usually not so high.

The structure of this paper is organized as follows: in Section 2 we first introduce the classical affine-type ALE mapping, then define the Piola-type ALE mapping and their relevant properties. Based on such novel ALE mapping and a proper stabilization technique, a new weak form of the parabolic/mixed parabolic moving interface problem is defined in Section 3. Then we proceed to define the semi-discrete novel ALE-FEM and analyze its stability and convergence theorems in Section 4, and define as well as analyze its fully discrete novel ALE-FEM in Section 5. Numerical experiments are carried out in Section 6 to validate all theoretical results. We end the paper with a few concluding remarks in Section 7.

Through out this paper, we adopt $C$ as a generic positive constant which is independent of all discretized parameters such as the mesh size $h$ and the time step size $\Delta t$. And, we set ${\hat{\psi }}_i={\hat{\psi }}_i({\hat{\mathit{x}}}_i,t)$ which equals ${\psi }_i({\mathit{x}}_i({\hat{\mathit{x}}}_i,t),t)$. For the simplicity of notations, we use $\nabla$ to denote the differentiation with respect to the current spatial variable $\mathit{x}$, and $\hat{\nabla }$ means to take derivatives with respect to the reference spatial variable $\hat{\mathit{x}}$, such as $\nabla {\psi }_i={\nabla }_{{\mathit{x}}_i}{\psi }_i$ and $\hat{\nabla }{\hat{\psi }}_i={\nabla }_{{\hat{\mathit{x}}}_i}{\hat{\psi }}_i(i=1,2)$, etc. In addition, we adopt the standard notation for Sobolev spaces $W^{l,p}\left(\Omega \right)$ and their associated norms and seminorms. For $p=2$, $W^{l,2}\left(\Omega \right)=H^l\left(\Omega \right)$. The standard $L^2$ inner product is adopted, as $(\psi ,\tilde{\psi })={\int }_{\Omega }\psi \tilde{\psi }d\mathit{x}$, $<\psi ,\tilde{\psi }{>}_{\partial \Omega }={\int }_{\partial \Omega }\psi \tilde{\psi }ds$. Some norm notations are given as ${\Vert {\psi }_i\Vert }_{0,{\Omega }_i^t}={\Vert {\psi }_i\Vert }_{L^2\left({\Omega }_i^t\right)}$, ${\Vert {\psi }_i\Vert }_{\infty ,{\Omega }_i^t}={\Vert {\psi }_i\Vert }_{L^{\infty }\left({\Omega }_i^t\right)}$, ${\Vert {\psi }_i\Vert }_{l,{\Omega }_i^t}={\Vert {\psi }_i\Vert }_{H^l\left({\Omega }_i^t\right)}(l>0,i=1,2)$.