Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction–diffusion problems

Abstract

We consider discretizations for reaction–diffusion systems with nonlinear diffusion in two space dimensions. The applied model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. We propose an implicit Voronoi finite volume discretization on arbitrary, even anisotropic, Voronoi meshes that allows to prove uniform, mesh-independent global upper and lower bounds for the chemical potentials. These bounds provide one of the main steps for a convergence analysis for the fully discretized nonlinear evolution problem. The fundamental ideas are energy estimates, a discrete Moser iteration and the use of discrete Gagliardo–Nirenberg inequalities.

Appendices

Appendix A: Discrete Gagliardo–Nirenberg inequality

In the following, we recapitulate from [1] a discrete version of the Gagliardo–Nirenberg inequality [40] and a discrete Sobolev–Poincaré inequality on admissible finite volume meshes [12]. The proof uses functions of bounded variation and is done under the general assumption on the mesh, that

$$\begin{aligned} \exists \xi >0:\quad \text {dist}(x_K,\sigma )\ge \xi \mathord {\left|x_K-x_L\right|}\quad \forall \sigma =K|L\in \mathcal {E}_{int}\cap \mathcal {E}_K\quad \forall x_K\in \mathcal {P}, \end{aligned}$$

(34)

see [1, Eq. (4)]. In the case of Voronoi meshes this condition holds true with $\xi =\frac{1}{2}$.

Lemma 5

(see [1, Th. 3 and Th. 4]) Let $\varOmega$ be an open bounded polyhedral domain of $\mathbb {R}^d$, $d\ge 2$. Let $\mathcal {M}=(\mathcal {P},\mathcal {V},\mathcal {E})$ be a given admissible finite volume mesh which satisfies (34).

• Then, there exists a constant $C>0$ only depending on $d$ and $\varOmega$ (but not on $\mathcal {M}$) such that

  $$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}\le C_{gn,p}\mathord {\left||w_h\right||}_{L^1}^{1-\theta }\mathord {\left||w_h\right||}_{H^1,\mathcal {M}}^{\theta },\quad C_{gn,p}\!:=\!\frac{C}{\xi ^{\theta /2}}, \quad \forall w_h\!\in \! X_{\mathcal {V}}(\mathcal {M}),\qquad \end{aligned}$$

  (35)

  where

  $$\begin{aligned} \theta =\frac{2d(1-p)}{(d+2)p},\quad p\in {\left\{ \begin{array}{ll} \left[ 1,\infty \right) &{} \text { if } d=2,\\ \left[ 1,2d/(d-2)\right] &{} \text { if } d>2. \end{array}\right. } \end{aligned}$$

  (36)

• Let $d\le 2$. Then for all $p\in [1,\infty )$ there exists a constant $C>0$ only depending on $p$, $d$ and $\varOmega$ (but not on $\mathcal {M}$) such that

  $$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}\le \frac{C}{\xi ^{1/2}}\mathord {\left||w_h\right||}_{H^1,\mathcal {M}} \quad \forall w_h\in X_{\mathcal {V}}(\mathcal {M}). \end{aligned}$$

  (37)

• Let $d> 2$. Then for all $1\le p \le \frac{2d}{d-2}$ there exists a constant $C>0$ only depending on $p$, $d$ and $\varOmega$ (but not on $\mathcal {M}$) such that

  $$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}\le \frac{C}{\xi ^{1/2}}\mathord {\left||w_h\right||}_{H^1,\mathcal {M}} \quad \forall w_h\in X_{\mathcal {V}}(\mathcal {M}). \end{aligned}$$

  (38)

Remark 8

An inspection of the proof shows that $C_{gn,p}$ depends continuously on $p$ and can therefore be estimated uniformly for $p\in [2,3]$. The results of [1, Th. 3 and Th. 4] allow us to handle all Voronoi meshes, including arbitrarily anisotropic grids. We remark that in the case of homogeneous Dirichlet boundary conditions a discrete Sobolev–Poincaré inequality on more general meshes than admissible meshes is proven in [13, Sec. 5].

In the proof of Theorem 3 we need a special case of the discrete Gagliardo–Nirenberg inequality:

Corollary 2

Let $\varOmega$ be an open bounded polygonal domain in $\mathbb {R}^2$. Let $\mathcal {M}=(\mathcal {P},\mathcal {V},\mathcal {E})$ be a given admissible finite volume mesh which satisfies (34). For any $\epsilon >0$ and any $p\in (1,3]$ there exists a constant $c_{\epsilon ,p}>0$ such that

$$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}^p\le \frac{\epsilon }{\xi ^{(p-1)/2}} \mathord {\left||w_h\ln \mathord {\left|w_h\right|}\right||}_{L^1} \mathord {\left||w_h\right||}_{H^1,\mathcal {M}}^{p-1}+ c_{\epsilon ,p} \mathord {\left||w_h\right||}_{L^1}. \end{aligned}$$

(39)

(For $\epsilon \rightarrow 0$ it follows $c_{\epsilon ,p}\rightarrow \infty$).

Proof

For $N>1$ we define the function

$$\begin{aligned} \chi (s):={\left\{ \begin{array}{ll} 0, &{}\text {for } \mathord {\left|s\right|}\le N,\\ 2(\mathord {\left|s\right|}-N), &{}\text {for } \mathord {\left|s\right|}\in (N, 2N], \\ \mathord {\left|s\right|}, &{}\text {for } \mathord {\left|s\right|}> 2 N. \end{array}\right. } \end{aligned}$$

Adding and subtracting $\chi (w_h)$ we obtain

$$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}^p \le c_p\left( \mathord {\left||\mathord {\left|w_h\right|}-\chi (w_h)\right||}_{L^p}^{p}+\mathord {\left||\chi (w_h)\right||}_{L^p}^p \right) . \end{aligned}$$

(40)

The first term of (40) can be estimated by

$$\begin{aligned} \mathord {\left||\mathord {\left|w_h\right|}-\chi (w_h)\right||}_{L^p}^{p}= \sum _{\begin{array}{c} K\in \mathcal {V}\\ \mathord {\left|w_K\right|}\le 2N \end{array}} \mathord {\left|K\right|} \left( \mathord {\left|w_K\right|}-\chi (w_K)\right) ^{p} \le (2N)^{p-1} \mathord {\left||w_h\right||}_{L^1}. \end{aligned}$$

Applying (35) for the second term of (40) we obtain

$$\begin{aligned} \mathord {\left||\chi (w_h)\right||}_{L^p}^p&\le \frac{C^p}{\xi ^{p\theta /2}} \mathord {\left||\chi (w_h)\right||}_{L^1}^{p(1-\theta )}\mathord {\left||\chi (w_h)\right||}_{H^1,\mathcal {M}}^{p\theta }\\&= \frac{C^p}{\xi ^{(p-1)/2}} \mathord {\left||\chi (w_h)\right||}_{L^1}\mathord {\left||\chi (w_h)\right||}_{H^1,\mathcal {M}}^{p-1}. \end{aligned}$$

We estimate

$$\begin{aligned} \mathord {\left||\chi (w_h)\right||}_{L^1}&\le \sum _{\begin{array}{c} K\in \mathcal {V}\\ \mathord {\left|w_K\right|}> N \end{array}} \mathord {\left|K\right|} \mathord {\left|w_K\right|} \le \frac{1}{\ln N} \sum _{\begin{array}{c} K\in \mathcal {V}\\ \mathord {\left|w_K\right|}> N \end{array}} \mathord {\left|K\right|} \mathord {\left|w_K\right|}\ln \mathord {\left|w_K\right|}\\&\le \frac{1}{\ln N} \mathord {\left||w_h\ln \mathord {\left|w_h\right|}\right||}_{L^1} \end{aligned}$$

and since $\mathord {\left|\frac{\chi (w_L)-\chi (w_K)}{w_L-w_K}\right|}\le 2$ and $\mathord {\left|\chi (w_h)\right|}\le \mathord {\left|w_h\right|}$ we deduce

$$\begin{aligned} \mathord {\left||\chi (w_h)\right||}_{H^1,\mathcal {M}}^2&= \mathord {\left||\chi (w_h)\right||}_{L^2}^2+\mathord {\left|\chi (w_h)\right|}_{H^1,\mathcal {M}}^2 \le \mathord {\left||w_h\right||}_{L^2}^2+4\mathord {\left|w_h\right|}_{H^1,\mathcal {M}}^2\\&\le 4\mathord {\left||w_h\right||}_{H^1,\mathcal {M}}^2. \end{aligned}$$

All together we find

$$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}^p \le c_{\epsilon }\mathord {\left||w_h\right||}_{L^1} +\frac{\epsilon }{\xi ^{(p-1)/2}}\mathord {\left||w_h\ln \mathord {\left|w_h\right|}\right||}_{L^1} \mathord {\left||w_h\right||}_{H^1,\mathcal {M}}^{p-1} \end{aligned}$$

with $c_{\epsilon ,p}=c_p(2 N)^{p-1}$ and $\epsilon =\frac{c_p C^p 2^{p-1}}{\ln N}$. $\square$

Appendix B: Technical lemmas

In this part we collect some auxiliary results, which we use in Sect. 3.4 and in Sect. 3.6.

Lemma 6

Let $x,y,p\in \mathbb {R}$, $x,y>0$.

• 1. For $p\ge 2$ the following inequalities hold:

  $$\begin{aligned} \frac{4(p-1)}{p^2}\left( x^{p/2}-y^{p/2}\right) ^2 \le (x-y)(x^{p-1}-y^{p-1})\le \left( x^{p/2}-y^{p/2}\right) ^2.\qquad \end{aligned}$$

  (41)

• 2. For $p\ge 1$, we have

  $$\begin{aligned} \frac{1}{p}(x^{p}-y^{p}) \le x^{p-1}(x-y). \end{aligned}$$

  (42)

• 3. Finally, for $p\ge 2$ the inequalities

  $$\begin{aligned} \frac{2}{p^2}(x^{p/2}-y^{p/2})^2\le (x^{p-2}+y^{p-2})(x-y)^2\le 2(x^{p/2}-y^{p/2})^2 \end{aligned}$$

  (43)

  are fulfilled.

Proof

• 1. For $z\ge 1$, we consider the function

  $$\begin{aligned} f(z)=(z-1)(z^{p-1}-1)-\frac{4(p-1)}{p^2}(z^{p/2}-1)^2. \end{aligned}$$

  The first and second derivatives of $f$ are given by

  $$\begin{aligned} \frac{d}{d z} f(z)&= \frac{(p-2)^2}{p} z^{p-1} - (p-1) z^{p-2} +\frac{4(p-1)}{p}z^{p/2-1} -1,\\ \frac{d^2}{d z^2} f(z)&= \frac{(p-2)(p-1)}{p} \left( ((p-2)z-p)z^{p-3}+2z^{p/2-2}\right) . \end{aligned}$$

  It is easy to see that $f(1)=0$, $f^{\prime }(1)=0$ and $f^{\prime \prime }(1)=0$. Further, we deduce from $f^{\prime \prime }(z)>0$ for $z>1$ that $f^{\prime }(z)>0$ and $f(z)>0$. With $z=x/y$, $x\ge y > 0$ we find

  $$\begin{aligned} 0\le f(x/y)= (x-y)(x^{p-1}-y^{p-1}) -\frac{4(p-1)}{p^2}\left( x^{p/2}-y^{p/2}\right) ^2 \end{aligned}$$

  and finally by using Muirhead’s inequality

  $$\begin{aligned} \left( x^{p/2}-y^{p/2}\right) ^2-(x-y)(x^{p-1}-y^{p-1})= x^{p-1}y+x y^{p-1}-2 x^{p/2} y^{p/2} \ge 0 \end{aligned}$$

  holds. The case $y=0$ is trivial.

• 2. For the second statement we consider for $z\ge 1$ the function

  $$\begin{aligned} f(z)=\frac{p-1}{p} z^{p}- z^{p-1}+ \frac{1}{p}. \end{aligned}$$

  Since $f(1)=0$, the first derivative of $f$

  $$\begin{aligned} \frac{d}{d z} f(z) = (p-1) z^{p-2}(z-1)\ge 0 \end{aligned}$$

  implies $f(z)\ge 0$. Setting $z=x/y$, $x\ge y > 0$ we find

  $$\begin{aligned} 0 \le y^p f(x/y)= \left( \frac{p-1}{p}\frac{x^p}{y^p} -\frac{x^{p-1}}{y^{p-1}} +\frac{1}{p}\right) y^p \!=\!x^{p-1}(x-y)-\frac{1}{p}(x^p-y^p). \end{aligned}$$

  For $x\le y$ it results

  $$\begin{aligned} \frac{1}{p}(x^p-y^p)\le (x^p-y^p) \le x^{p-1}(x-y x^{-p+1}) \le x^{p-1}(x-y). \end{aligned}$$

• 3. Now let

  $$\begin{aligned} f(z)=(z^{p-2}+1)(z-1)^2-\frac{2}{p^2}(z^{p/2}-1)^2. \end{aligned}$$

  The first and second derivatives are given by

  $$\begin{aligned} \frac{d}{d z} f(z)&= 2(z-1)(z^{p-2}+1)+(p-2)(z-1)^2 z^{p-3} -\frac{2}{p}(z^{p/2}-1)z^{p/2-1},\\ \frac{d^2}{d z^2} f(z)&= \frac{1}{p}\left( 2p+(p-2)z^{p/2-2}+z^{p-4} g(z)\right) \end{aligned}$$

  with

  $$\begin{aligned} g(z)&= (p-3)(p-2)p-2(p-2)(p-1)p z+(p-1)(p^2-2)z^2,\\ g^{\prime }(z)&= 2(p-1)\left( (p^2-2) z-(p-2) p\right) ,\\ g^{\prime \prime }(z)&= 2(p-1)(p^2-2). \end{aligned}$$

  From $g^{\prime \prime }(z)>0$ for $z>1$ and $p\ge 2$ we see that $g(z)$ is a convex function. Furthermore $f(z)$ is convex since $g^{\prime }(1)=4(p-1)^2>0$, $g(1)=2$ and $f^{\prime \prime }(z)>0$. Using $f^{\prime }(1)=f(1)=0$ we get $f(z)>0$ and with $z=x/y$, $x\ge y > 0$ the first inequality of (43). The last assertion follows from

  $$\begin{aligned}&(x^{p/2}-y^{p/2})^2-\frac{1}{2}(x^{p-2}+y^{p-2})(x-y)^2\\&\quad = \frac{1}{2}\left( x^p+y^p-x^{p-2}y^2-x^2 y^{p-2}\right) +\left( x^{p-1}y+xy^{p-1}-2x^{p/2}y^{p/2}\right) \end{aligned}$$

  together with Muirhead’s inequality for the term

  $$\begin{aligned} x^p+y^p&\ge x^{p-2}y^2+x^2y^{p-2},&x^{p-1}y+xy^{p-1}&\ge 2 x^{p/2}y^{p/2}. \end{aligned}$$

$\square$

Lemma 7

Let $x$ be a real number. Then

$$\begin{aligned} f(x)=\frac{({{\mathrm{e}}}^{x}-1)({{\mathrm{e}}}^{-x}+1)}{2 x}&\ge 1,&g(x)=\frac{({{\mathrm{e}}}^{x}-1)({{\mathrm{e}}}^{-x}-1)}{x^2}&\le -1 \end{aligned}$$

hold. We define the value of the functions at $x=0$ as limit $x\rightarrow 0$.

Proof

Using the power series of $\sinh (x)$ and $\cosh (x)$ we obtain

$$\begin{aligned} f(x)&=1+\sum _{i=1}^{\infty } \frac{x^{2i}}{(2i+1)!},&g(x)=-1-2\sum _{i=2}^{\infty } \frac{x^{2i-2}}{(2 i)!}. \end{aligned}$$

$\square$