A convergent FEM-DG method for the compressible Navier–Stokes equations

Abstract

This paper presents a new numerical method for the compressible Navier–Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix–Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax–Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.

Existence of a numerical solution

Since the numerical method in Definition 3.1 is nonlinear and implicit it is not trivial that it is actually well-defined (i.e admits a solution). In addition, the discretization of the momentum transport is posed using element averages of the velocity. As we will see the only part of the discretization that provides sufficient number of equations to determine all the degrees of freedom of $u_h$ is the discretization of the diffusion operator. Hence, in it’s present form, our discretization is not suitable for the Euler equations.

The purpose of this section, is to prove the following the existence result which we have relied on in our analysis.

Proposition 3.3.

For each fixed $h > 0$, there exists a solution

$$\begin{aligned} (\varrho _h^k, u_h^k) \in Q_h({\Omega })\times V_h({\Omega }),\quad \varrho ^k_h(\cdot ) > 0, \quad k=1, \ldots , M, \end{aligned}$$

to the numerical method posed in Definition 3.1.

To prove this result, we shall use a topological degree argument. The argument is strongly inspired by a very similar argument in the paper [12]. We will argue the existence of solutions to the following finite element map.

Definition 11.1

Let the finite element map

$H:Q_h^+({\Omega }) \times V_h({\Omega }) \times [0,1] ~\mapsto ~ Q_h({\Omega }) \times V_h({\Omega })$ be given by

$$\begin{aligned} H(\varrho _h, u_h, \alpha ) = (f_h(\alpha ), g_h(\alpha )), \end{aligned}$$

where $(f_h(\alpha ), g_h(\alpha ))$ are obtained through the mappings:

$$\begin{aligned} \int _{\Omega }f_h(\alpha )q_h ~dx&= \int _{\Omega }\frac{\varrho _h - \varrho _h^{k-1}}{\Delta t} q_h~dx \nonumber \\&-~\alpha \sum _\Gamma \int _{\Gamma } \text{ Up}(\varrho u)[\![q_h]\!]_\Gamma ~dS(x) \nonumber \\&+~\alpha h^{1-\epsilon }\sum _\Gamma \int _{\Gamma }[\![\varrho _h]\!]_\Gamma [\![q_h]\!]_\Gamma ~dS(x), \end{aligned}$$

(11.1)

for all $q_h \in Q_h({\Omega })$ and

$$\begin{aligned} \int _{\Omega }g_h(\alpha )v_h~dx&= \int _{\Omega }\frac{\varrho _h\widehat{u}_h - \varrho _h^{k-1}\widehat{u}^{k-1}_h}{\Delta t} v_h~dx + \int _{\Omega }\nabla _h u_h: \nabla _h v_h~dx \nonumber \\&-~\alpha \sum _{ \Gamma } \int _{\Gamma } \text{ Up}(\varrho u\otimes \widehat{u})[\![\widehat{v}]\!]_\Gamma ~dS(x) \nonumber \\&-~\alpha \int _{\Omega }p(\varrho _h)\text{ div}\,v_h~dx \nonumber \\&-~\alpha h^{1-\epsilon }\sum _E \int _{\partial E} \left( \frac{\widehat{u}_- + \widehat{u}_+}{2}\right) [\![\varrho _h]\!]\widehat{v}_h~dS(x), \end{aligned}$$

(11.2)

for all $v_h \in V_h({\Omega })$.

Observe that a solution of $H(\varrho _h, u_h, 1) = (0,0)$ is a solution to our numerical method as posed in Definition 3.1.

Before proceeding, let us make clear what we mean by topological degree in the present finite element context and denote by $d(F,{\Omega },y)$ the $\mathbb Z$–valued (Brouwer) degree of a continuous function $F:\bar{O} \rightarrow \mathbb R ^M$ at a point $y\in \mathbb R ^M\backslash F(\partial O)$ relative to an open and bounded set $O \subset \mathbb R ^M$.

Definition 11.2

Let $S_{h}$ be a finite element space, $\Vert \cdot \Vert$ be a norm on this space, and introduce the bounded set

$$\begin{aligned} \tilde{S}_{h} = \left\{ q_{h} \in S_{h}; \Vert q_{h}\Vert \le C\right\} , \end{aligned}$$

where $C>0$ is a constant. Let $\{\sigma _{i}\}_{i=1}^M$ be a basis such that $\text{ span}\{\sigma _{i}\}_{i=1}^M = S_{h}$ and define the operator $\Pi _\mathcal{B }:S_{h} \rightarrow \mathbb R ^M$ by

$$\begin{aligned} \Pi _\mathcal{B }q_{h} = (q_{1},q_{2}, \ldots , q_{M}), \quad \quad q_{h} = \sum _{i=1}^M q_{i}\sigma _{i}. \end{aligned}$$

The degree $d_{S_{h}}(F,\tilde{S}_{h},q_{h})$ of a continuous mapping $F:\tilde{S}_{h} \rightarrow S_{h}$ at $q_{h} \in S_{h}\backslash F(\partial \tilde{S}_h)$ relative to $\tilde{S}_{h}$ is defined as

$$\begin{aligned} d_{S_{h}}(F,\tilde{S}_{h},q_{h}) = d\left( \Pi _\mathcal{B } F(\Pi _\mathcal{B }^{-1}), \Pi _\mathcal{B } \tilde{S}_{h}, \Pi _\mathcal{B }q_{h}\right) . \end{aligned}$$

The next lemma is a consequence of some basic properties of the degree, cf. [7].

Lemma 11.3

Fix a finite element space $S_{h}$, and let $d_{S_{h}}(F, \tilde{S}_{h},q_{h})$ be the associated degree of Definition 11.2. The following properties hold:

• (1) $d_{S_{h}}(F,\tilde{S}_{h},q_{h})$ does not depend on the choice of basis for $S_{h}$.

• (2) $d_{S_{h}}(\mathrm Id ,\tilde{S}_{h}, q_{h}) = 1$.

• (3) $d_{S_{h}}(H(\cdot ,\alpha ),\tilde{S}_{h},q_{h}(\alpha ))$ is independent of $\alpha \in J:=[0,1]$ for $H\!:\!\tilde{S}_{h}\times J \rightarrow S_{h}$ continuous, $q_h:J \rightarrow S_{h}$ continuous, and $q_{h}(\alpha ) \notin H(\partial \tilde{S}_{h},\alpha )$ $\forall \alpha \in [0,1]$.

• (4) $d_{S_{h}}(F,\tilde{S}_{h},q_{h}) \ne 0 \Rightarrow F^{-1}(q_{h}) \ne \emptyset$.

To prove Proposition 3.3, we shall apply Lemma 11.3 with $q_h = 0$ and mapping $H$ given by Definition 11.1. Let us first prove that our mapping $H$ satisfies (3) in Lemma 11.3.

Lemma 11.4

Let $H:Q_h^+({\Omega }) \times V_h({\Omega }) \times [0,1] ~\mapsto ~ Q_h({\Omega }) \times V_h({\Omega })$ be the finite element mapping of Definition 11.1. There is a subset $\tilde{S}_h \subset Q_h^+({\Omega }) \times V_h({\Omega })$ for which $H\!:\!\tilde{S}_{h}\times J \rightarrow S_{h}$ is continuous and the zero solution $(0,0) \not \in H(\partial \tilde{S}, \alpha )$ for all $\alpha \in [0, 1]$.

Proof

For any subset $\tilde{S} \subset Q_h^+({\Omega }) \times V_h({\Omega })$ bounded independently of $\alpha$, the corresponding mapping $H(\tilde{S}_h, \alpha , 0)$ is clearly continuous. This follows directly from (11.1) and (11.2) using the equivalence of finite dimensional norms. The more involved part is to determine a subset $\tilde{S}_h$ for which $(0,0) \not \in H(\partial \tilde{S}, \alpha )$ independently of $\alpha$.

Now, let us for the moment assert the existence of of $(\varrho , u)$ satisfying

$$\begin{aligned} H(\varrho _h, u_h, \alpha ) = (0, 0). \end{aligned}$$

Then, from Lemma 3.4, we have that $\varrho _h > 0$ and moreover (11.1) yields

$$\begin{aligned} \int _{\Omega }\varrho _h~dx = \int _{\Omega }\varrho _h^{k-1}~dx. \end{aligned}$$

Consequently, we can conclude that

$$\begin{aligned} \Vert \varrho _h\Vert _{L^\infty ({\Omega })} \le C_\dagger , \end{aligned}$$

(11.3)

independently of $\alpha$.

To derive a bound on the velocity $u_h$, we can repeat the steps in the proof of Proposition 4.2 (the energy estimate) while keeping track of $\alpha$ to obtain

$$\begin{aligned}&\int _{\Omega }\frac{\varrho _h |\widehat{u}_h|^2}{2} + \frac{\alpha }{\gamma -1}p(\varrho _h)~dx + \Delta t\int _{\Omega }|\nabla u_h|^2~dx \\&\quad \quad \le \int _{\Omega }\frac{\varrho ^{k-1}_h |\widehat{u}^{k-1}_h|^2}{2} + \frac{\alpha }{\gamma -1}p(\varrho ^{k-1}_h)~dx \\&\quad \quad \le \int _{\Omega }\frac{\varrho ^{k-1}_h |\widehat{u}^{k-1}_h|^2}{2} + \frac{1}{\gamma -1}p(\varrho ^{k-1}_h)~dx \le C, \end{aligned}$$

where $C$ is independent of $\alpha$. Together with (11.3), this allow us to conclude

$$\begin{aligned} \Vert \varrho _h\Vert _{L^\infty ({\Omega })} + \Vert u_h\Vert _{L^\infty ({\Omega })} \le C_\dagger . \end{aligned}$$

We can now define the subspace

$$\begin{aligned} \tilde{S}_h = \left\{ (\varrho _h, u_h) \in Q_h^+\times V_h; \Vert \varrho _h\Vert _{L^\infty ({\Omega })} + \Vert u_h\Vert _{L^\infty ({\Omega })} \le C_\dagger \right\} , \end{aligned}$$

which by definition has the property that $(0,0) \not \in H(\partial \tilde{S}, \alpha )$ for all $\alpha \in [0,1]$. This concludes the proof. $\square$

Lemma 11.5

Let $\tilde{S}$ be the subspace obtained by the previous lemma. Then, the topological degree of $H(\tilde{S}_h, 0)$ at $q_h = 0$ is non-zero:

$$\begin{aligned} d_{S_h}(H(\cdot , 0), \tilde{S}_h, 0) \ne 0. \end{aligned}$$

(11.4)

As a consequence, there exists $(\varrho _h, u_h) \in \tilde{S}_h$ such that

$$\begin{aligned} H(\varrho _h, u_h, 1) = (0,0), \end{aligned}$$

and hence Proposition 3.3 holds true.

Proof

First, we note that proving (11.4) is equivalent to proving the existence of $(\varrho _h, u_h) \in Q^+_h \times V_h$ satisfying, for all $(q_h, v_h) \in Q_h\times V_h$,

$$\begin{aligned}&\int _{\Omega }\varrho _h q_h~dx = \int _{\Omega }\varrho ^{k-1}_h q_h~dx, \nonumber \\&\int _{\Omega }\varrho _h \widehat{u}_h v_h~dx + \Delta t \int _{\Omega }\nabla _h u_h: \nabla _h v_h~dx = \int _{\Omega }\varrho _h^{k-1}\widehat{u}_h^{k-1} v_h~dx. \end{aligned}$$

(11.5)

The first equation has the solution $\varrho _h = \varrho _h^{k-1}$. Setting this into the second equation in (11.5), we see that the resulting linear system is a sum of a positive matrix $\varrho ^{k-1}_h \widehat{u}_h v_h$ and a symmetric positive definite matrix $\Delta t\nabla _h u_h \nabla _h v_h$. Since the Laplace problem with the Crouzeix–Raviart element space and dirichlet conditions is well-defined, there is no problems with concluding the existence of $u_h$ satisfying the second equation in (11.5). $\square$