Nonsmooth exact penalization second-order methods for incompressible bi-viscous fluids

Abstract

We consider the exact penalization of the incompressibility condition $\text {div}(\mathbf {u})=0$ for the velocity field of a bi-viscous fluid in terms of the $L^1$–norm. This penalization procedure results in a nonsmooth optimization problem for which we propose an algorithm using generalized second-order information. Our method solves the resulting nonsmooth problem by considering the steepest descent direction and extra generalized second-order information associated to the nonsmooth term. This method has the advantage that the divergence-free property is enforced by the descent direction proposed by the method without the need of build-in divergence-free approximation schemes. The inexact penalization approach, given by the $L^2$-norm, is also considered in our discussion and comparison.

Appendix

Lemma 8

Let $\gamma$ and $\sigma$ be two positive constants. The function $\phi :\mathbb {R}\rightarrow \mathbb {R}$ defined by $\phi (a):=\gamma \sigma \frac{ a}{\max (\sigma , \gamma |a|)}$ is Lipschitz continuous and semismooth.

Proof

Let us start by rewriting $\phi (a)$ as $\phi (a)=\gamma \sigma \frac{a}{\phi _m(a)}$, with $\phi _m(a):=\max (\sigma ,\gamma |a|)$. Next, we notice that the max function is globally Lipschitz continuous with constant $L_{max}$. This fact implies that

$$\begin{aligned} \begin{array}{lll} |\phi _m(a_1)-\phi _m(a_2)|= |\max (\sigma ,\gamma |a_1|)- \max (\sigma ,\gamma |a_2|)|\vspace{0.2cm}\\ \le \gamma L_{max}||a_1|-|a_2||\le \gamma L_{max}|a_1-a_2|,\,\, \forall a_1,a_2\in \mathbb {R}. \end{array} \end{aligned}$$

(78)

We conclude that $\phi _m$ is Lipschitz continuous. Considering this result, we have that

$$\begin{aligned} \begin{array}{lll} |\phi (a_1)-\phi (a_2)| &{}=&{} \left| \gamma \sigma \frac{a_1}{\phi _m(a_1)} -\gamma \sigma \frac{a_2}{\phi _m(a_2)}\right| \vspace{0.2cm}\\ {} &{}=&{} \gamma \sigma \left| \frac{a_1}{\phi _m(a_1)}-\frac{a_2}{\phi _m(a_2)} +\frac{a_1}{\phi _m(a_2)}- \frac{a_1}{\phi _m(a_2)}\right| \vspace{0.2cm}\\ {} &{} \le &{} \gamma \sigma \left| a_1\left( \frac{\phi _m(a_2)- \phi _m(a_1)}{\phi _m(a_1)\phi _m(a_2)}\right) \right| +\gamma \sigma \left| \frac{1}{\phi _m(a_2)} (a_1 - a_2)\right| . \end{array} \end{aligned}$$

Now, it is clear that $0<\sigma \le \phi _m(a_2)$, thus $\frac{1}{\phi _m(a_2)}\le \frac{1}{\sigma }$. By plugging this inequality in the above expression, we have that

$$\begin{aligned} \begin{array}{lll} |\phi (a_1)-\phi (a_2)| &{}\le &{} \gamma \sigma \left| \frac{a_1}{\phi _m(a_1)}\left( \frac{\phi _m(a_2)- \phi _m(a_1)}{\sigma }\right) \right| +\gamma |a_1 - a_2|\vspace{0.2cm}\\ &{}=&{} \gamma \left| \frac{a_1}{\phi _m(a_1)}\right| \left| \phi _m(a_2)- \phi _m(a_1)\right| + \gamma |a_1-a_2|. \end{array} \end{aligned}$$

Finally, since $\left| \frac{a_1}{\phi _m(a_1)}\right| \le \frac{1}{\gamma }$, we conclude, thanks to (78), that

$$\begin{aligned} |\phi (a_1)-\phi (a_2)| \le \gamma (L_{max} +1)|a_1-a_2|. \end{aligned}$$

Regarding the semismoothness of $\phi$, note that the absolute value $|\cdot |: \mathbb {R} \rightarrow \mathbb {R}$ and the function $\max (0, \cdot ) : \mathbb {R} \rightarrow \mathbb {R}$ are both semismooth (see [39, Sect. 2.5] and [28, Lemma 3.1] respectively). Then, since the composition of semismooth functions in $\mathbb {R}^n$ is a semismooth function [39, Prop. 2.9], it follows that $\phi (a)$ is semismooth.

Remark 4

The function $\varphi _j:\mathbb {R}^m \rightarrow \mathbb {R}$ defined by $\varphi _j(a):=\gamma \sigma \frac{ a_j}{\max (g, \beta |a|)}$ is also Lipschitz continuous and semismooth. The proof of this assertion is analogous to the one given in Lemma 8.

Lemma 9

Let $\phi ({{\,\mathrm{div}\,}}\mathbf {u}(x))=\displaystyle \sigma \gamma \frac{ {{\,\mathrm{div}\,}}\mathbf {u}(x)}{\max (\sigma , \gamma |{{\,\mathrm{div}\,}}\mathbf {u}(x)|)}$ with $\gamma$ and $\sigma$ positive constants. A measurable selection $M_{\phi }( \mathbf {u})$ of Clarke’s generalized Jacobian $\partial \phi ({{\,\mathrm{div}\,}}\mathbf {u})$ is :

$$\begin{aligned} M_{\phi }( \mathbf {u}(x))={\left\{ \begin{array}{ll} \displaystyle \sigma \frac{1}{|{{\,\mathrm{div}\,}}\mathbf {u}(x)|} \, - \sigma \displaystyle \frac{({{\,\mathrm{div}\,}}\mathbf {u}(x) {{\,\mathrm{div}\,}}\mathbf {u}(x)) }{|{{\,\mathrm{div}\,}}\mathbf {u}(x)|^3}, &{} if \,\, \gamma |{{\,\mathrm{div}\,}}\mathbf {u}(x)| \ge \sigma \vspace{0.2cm} \\ \gamma , &{} if \,\, \gamma |{{\,\mathrm{div}\,}}\mathbf {u}(x)| < \sigma . \end{array}\right. } \end{aligned}$$

(79)

a.e on $\Omega$

Proof

Let $\phi _3=\phi _1 \circ \phi _2$, where $\phi _1(z)= \max (0,z) + \sigma$ and $\phi _2(y)= \gamma |y| - \sigma$. Then the following identity holds:

$$\begin{aligned} \phi _3(y)=\max (\sigma , \gamma |y|)= \max (0, \gamma | y| -\sigma ) + \sigma . \end{aligned}$$

From [28, pp. 869] we have that $M_{\phi _1}( \gamma |y| - \sigma ) \in \partial \phi _1(\gamma |y| - \sigma )$ given by

$$\begin{aligned} M_{\phi _1}( \gamma |y| - \sigma )={\left\{ \begin{array}{ll} 1 , &{} if \,\, \gamma |y| - \sigma >0\\ 0, &{} if \,\, \gamma |y| - \sigma \le 0, \end{array}\right. } \end{aligned}$$

is a measurable selection of $\partial \phi _1(\gamma |y| - \sigma )$. Next, since $\phi _2$ involves the function $| \cdot |$ evaluated at $y\ne 0$. From [39, Exaple 2.5.1] we have that

$$\begin{aligned} M_{\phi _2}(y)\in \partial \phi _2(y)=\displaystyle \bigg \{ \frac{\gamma y }{|y|}\bigg \} \, \text {for } y \ne 0 . \end{aligned}$$

Moreover, the chain rule for Clarke’s generalized Jacobian [39, Prop. 2.3] yields that:

$$\begin{aligned} M_{\phi _3}(y) v \in \partial \phi _3(y) v \subset co\{ M_{\phi _1} M_{\phi _2} v: M_{\phi _1} \in \partial \phi _1( \phi _2(y)) , M_{\phi _2} \in \partial \phi _2(y)\}. \end{aligned}$$

Thus, since $y \ne 0$,

$$\begin{aligned} M_{\phi _3}(y)={\left\{ \begin{array}{ll} \frac{\gamma y }{|y|} , &{} if \,\, \gamma |y| - \sigma >0\\ 0, &{} if \,\, \gamma |y| - \sigma \le 0, \end{array}\right. } \end{aligned}$$

(80)

Clearly, $\phi (y)= \sigma \gamma \frac{y}{ \phi _3(y)}$. Then, from the composition of functions we obtain that

$$\begin{aligned} M_{\phi }(y) \in \partial \phi (y) \subseteq \sigma \gamma \frac{\phi _3(y) \cdot 1 - y \partial \phi _3(y)\,\, }{\phi _3(y)^2}. \end{aligned}$$

Then, from (80) the following cases can occur:

• $\gamma | y| > \sigma$. Then:

  $$\begin{aligned} M_{\phi }(y) = \displaystyle \sigma \frac{1}{|y|} \, - \sigma \displaystyle \frac{y^2}{|y|^3} = 0. \end{aligned}$$

• $\gamma | y| \le \sigma$ gives:

  $$\begin{aligned} M_{\phi }(y) = \gamma . \end{aligned}$$

Finally, by taking $y={{\,\mathrm{div}\,}}\mathbf {u}(x)$ we have the desired result.

Remark 5

The measurable selection $N_j( \mathbf { u}(x))$ of Clarke’s generalized Jacobian $\partial \varphi _j(\mathcal {E} \mathbf {u}(x))$ is obtained by an analogous procedure to Lemma 9.