An iterative domain decomposition method for free boundary problems with nonlinear flux jump constraint

Abstract

In this paper we design an iterative domain decomposition method for free boundary problems with nonlinear flux jump condition. The proposed scheme requires, in each iteration, the approximation of the flux on (both sides of) the free interface. We present a finite element implementation of our method. The numerical implementation uses harmonically deformed triangulations to inexpensively generate finite element meshes in subdomains. We apply our method to a simplified model for jet flows in pipes and to a magnetohydrodynamics model. Finally, we present numerical examples illustrating the robustness and convergence of our scheme.

Appendix: An approximation of the flux

Given a free interface approximation $\Gamma ^h$, we consider the approximation of the flux of $u^h$ [the solution of problem (11)] on ${\Omega ^+_{h}}$.

Denote by $A=[a_{ij}]$ the Neumann finite element matrix defined by

$$\begin{aligned} a_{ij}=\int _{\Omega ^+_h}a_1\nabla \phi _{i}\nabla \phi _{j}\;\mathrm{d}x \end{aligned}$$

where $\{\phi _i\}$ are the usual hat basis function of the space ${\mathcal P}^1(\mathcal {T}^h_n, \Omega ^+_{h})$.

We classify the nodes in interior nodes I, boundary nodes $\Sigma ^+$ and interface nodes $\Gamma$. This classification gives the following block structure of the matrix $A$:

$$\begin{aligned} A= \left( \begin{array}{l@{\quad }l@{\quad }l} A_{II} &{} A_{I\Sigma ^+} &{} A_{I\Gamma }\\ A_{I\Sigma ^+}^\mathrm{T} &{} A_{\Sigma ^+\Sigma ^+} &{} A_{\Sigma ^+\Gamma }\\ A_{I\Gamma }^\mathrm{T} &{} A_{\Sigma ^+\Gamma }^\mathrm{T} &{} A_{\Gamma \Gamma }\\ \end{array} \right) . \end{aligned}$$

The solution of (11) is given by:

$$\begin{aligned} u^h= \left( \begin{array}{l} u_{I}^h\\ u_{\Sigma ^+}^h\\ u_{\Gamma }^h\\ \end{array} \right) = \left( \begin{array}{l} A_{II}^{-1}(A_{I\Sigma ^+}g^h)\\ g^h\\ 0\\ \end{array} \right) . \end{aligned}$$

We define $\mu$ by

$$\begin{aligned} \mu =A_{I\Gamma }^\mathrm{T} u_{I}=A_{I\Gamma }^\mathrm{T}A_{II}^{-1}A_{I\Sigma ^+}g^h. \end{aligned}$$

Let $N_\Gamma$ be the number of vertices of ${\mathcal T}^h$ on $\Gamma ^h$. We note that using basic finite element analysis $\mu =(\mu _{i}) \in \mathbb {R}^{N_\Gamma }$ with

$$\begin{aligned} \mu _{i} = \int _{\Omega ^+_{h}} (a_1 \nabla u^h ) \cdot \nabla \phi _{\ell _i} \;\mathrm{d}x = \int _{\Gamma ^h} (a_1 \nabla u^h )\cdot \eta _{\Gamma ^h} \phi _{\ell _i} \;\mathrm{d}s. \end{aligned}$$

Here, given $i\in \{1,\ldots ,N_\Gamma \}$, $\ell _i$ represents the index of the a node of $\mathcal {T}^h$ belonging to $\Gamma ^h$.

We use $\mu$ to obtain a piecewise linear approximation of the flux $\nabla u^h \cdot \eta _{\Gamma ^h}$. Since $u^h=0$ on $\Gamma ^h$, for each edge of $e_k$ of $\Gamma ^h$ we have

$$\begin{aligned} \nabla u^h|_{e_k}= \partial _{\eta _k} u \eta _k \end{aligned}$$

where $\eta _k$ represents the normal vector to edge $e_k$ pointing in the outward direction of $\Omega ^+_{h}$. Hence,

$$\begin{aligned} \mu _{i} = \int _{\Gamma ^h} (\eta _{\Gamma ^h}^\mathrm{T} a_1 \eta _{\Gamma ^h}) \partial _{\eta _{\Gamma ^h}} u \phi _{\ell _i} \;\mathrm{d}s. \end{aligned}$$

(33)

We define $\lambda ^h_1$ the piecewise linear approximation of $\partial _{\eta _{\Gamma ^h}} u$ as follows. First, we observe that $\lambda ^h_1 \in \mathrm{span} \{ \phi _{\ell _i}|_{\Gamma ^h}\}_{1 \le i \le N_\Gamma }$. Next, we introduce the matrix $Q=[q_{ij}] \in \mathbb {R}^{N_\Gamma \times N_\Gamma }$ with

$$\begin{aligned} q_{i j}=\int _{\Gamma ^h} (\eta _{\Gamma ^h}^\mathrm{T} a_1 \eta _{\Gamma ^h}) \phi _{\ell _i}\phi _{\ell _j} \;\mathrm{d}s. \end{aligned}$$

Finally, based on relation (33) we define

$$\begin{aligned} \lambda ^h_1= \sum _{i}^{N_\Gamma } \alpha _i \phi _{\ell _i}|_{\Gamma ^h} \end{aligned}$$

(34)

where $\alpha =(\alpha _i)$ is the solution of

$$\begin{aligned} Q \alpha =\mu . \end{aligned}$$

In a similar way, we define $\lambda _2^h$, the approximation of of the flux on $\Gamma ^h$, of the solution of (12).

Remark 2

A more regular approximation of the flux can be done in practice. For instance, we could obtain $\alpha$ as the solution of the following problem:

$$\begin{aligned} (Q +\epsilon D) \alpha =\mu . \end{aligned}$$

where $D$ is diffusion of operator on $\Gamma ^h$ and $\epsilon$ is a regularization parameter.