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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    Bessemoulin-Chatard, M., Chainais-Hillairet, C., Filbet, F.: On discrete functional inequalities for some finite volume schemes (April 10, 2013). http://math.univ-lyon1.fr/filbet/Papers/paper34.pdf. Submitted

  2. 2.

    Bothe, D., Pierre, M.: Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate. J. Math. Anal. Appl. 368(1), 120–132 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  3. 3.

    Brown, A.J.: Enzym action. J. Chem. Soc. 81, 373–386 (1902)

    Article  Google Scholar 

  4. 4.

    Carberry, J.: Chemical and catalytic reaction engineering. Dover Books on Chemistry Series. Dover Publications, Dover (2001)

  5. 5.

    Chou, S.H., Tang, S.: Conservative \(P1\) conforming and nonconforming Galerkin FEMs: effective flux evaluation via a nonmixed method approach. SIAM J. Numer. Anal. 38(2), 660–680 (2000)

    MATH  MathSciNet  Article  Google Scholar 

  6. 6.

    Desvillettes, L., Fellner, K.: Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations. J. Math. Anal. Appl. 319(1), 157–176 (2006)

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Desvillettes, L., Fellner, K., Pierre, M., Vovelle, J.: Global existence for quadratic systems of reaction-diffusion. Adv Nonlinear Stud 7(3), 491–511 (2007)

    MATH  MathSciNet  Google Scholar 

  8. 8.

    Deuflhard, P., Bornemann, F.: Gewöhnliche Differentialgleichungen. No. Bd. 2 in Numerische Mathematik. de Gruyter, Germany (2008)

  9. 9.

    Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

  10. 10.

    Erdmann, A., Shao, F., Fuhrmann, J., Fiebach, A., Patis, G.P., Trefonas, P.: Modeling of double patterning interactions in litho-cure-litho-etch (lcle) processes. p. 76400B. SPIE (2010)

  11. 11.

    Eymard, R., Fuhrmann, J., Gärtner, K.: A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems. Numer. Math. 102(3), 463–495 (2006)

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Eymard, R., Gallouët, T., Herbin, R.: The finite volume method. In: Ciarlet, P., Lions, J.L. (eds.) Handbook of Numerical Analysis, pp. 713–1020. North Holland, Amsterdam (2000)

  13. 13.

    Eymard, R., Gallouët, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes sushi: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30(4), 1009–1043 (2010). doi:10.1093/imanum/drn084

  14. 14.

    Eymard, R., Hilhorst, D., Murakawa, H., Olech, M.: Numerical approximation of a reaction-diffusion system with fast reversible reaction. Chin. Ann. Math. Ser. B 31(5), 631–654 (2010). doi:10.1007/s11401-010-0604-5

    Google Scholar 

  15. 15.

    Fiebach, A.: A dissipative finite volume scheme for reaction-diffusion systems in heterogeneous materials. Ph.D. thesis, Freie Universität Berlin (Submitted December 2013)

  16. 16.

    Fuhrmann, J., Fiebach, A., Patsis, G.P.: Macroscopic and stochastic modeling approaches to pattern doubling by acid catalyzed cross-linking. In: Proceedings of SPIE, vol. 7639, pp. 76392I. SPIE (2010)

  17. 17.

    Fuhrmann, J., Fiebach, A., Uhle, M., Erdmann, A., Szmanda, C.R., Truong, C.: A model of self-limiting residual acid diffusion for pattern doubling. Microelectron. Eng. 86(4–6), 792–795 (2009)

    Article  Google Scholar 

  18. 18.

    Fuhrmann, J., Linke, A., Langmach, H.: A numerical method for mass conservative coupling between fluid flow and solute transport. Appl. Numer. Math. 61(4), 530–553 (2011)

    MATH  MathSciNet  Article  Google Scholar 

  19. 19.

    Gajewski, H., Gröger, K.: Reaction–diffusion processes of electrically charged species. Math. Nachr. 177(1), 109–130 (1996)

    MATH  MathSciNet  Article  Google Scholar 

  20. 20.

    Gajewski, H., Skrypnik, I.V.: Existence and uniqueness results for reaction-diffusion processes of electrically charged species. Nonlinear Elliptic and Parabolic Problems (Zurich 2004). Progress in Nonlinear Differential Equations and Their Applications, vol. 64, pp. 151–188. Birkhäuser, Basel (2005)

  21. 21.

    Gärtner, K.: Existence of bounded discrete steady-state solutions of the van Roosbroeck system on boundary conforming Delaunay grids. SIAM J. Sci. Comput. 31(2), 1347–1362 (2009)

    MATH  Article  Google Scholar 

  22. 22.

    Giovangigli, V.: Multicomponent flow modeling. Modeling and Simulation in Science, Engineering & Technology. Birkhäuser, Boston (1999)

  23. 23.

    Glitzky, A.: Exponential decay of the free energy for discretized electro-reaction-diffusion systems. Nonlinearity 21(9), 1989–2009 (2008)

    MATH  MathSciNet  Article  Google Scholar 

  24. 24.

    Glitzky, A.: Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems. Math. Nachr. 284(17–18), 2159–2174 (2011)

    MATH  MathSciNet  Article  Google Scholar 

  25. 25.

    Glitzky, A., Gärtner, K.: Energy estimates for continuous and discretized electro-reaction-diffusion systems. Nonlinear Anal. Theory Methods Appl. Ser. A 70(2), 788–805 (2009)

    MATH  Article  Google Scholar 

  26. 26.

    Glitzky, A., Griepentrog, J.A.: Discrete Sobolev–Poincaré inequalities for Voronoi finite volume approximations. SIAM J. Numer. Anal. 48(1), 372–391 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  27. 27.

    Glitzky, A., Gröger, K., Hünlich, R.: Free energy and dissipation rate for reaction diffusion processes of electrically charged species. Appl. Anal. 60(3–4), 201–217 (1996)

    MATH  MathSciNet  Article  Google Scholar 

  28. 28.

    Glitzky, A., Hünlich, R.: Energetic estimates and asymptotics for electro-reaction-diffusion systems. Z. Angew. Math. Mech. 77(11), 823–832 (1997)

    MATH  MathSciNet  Article  Google Scholar 

  29. 29.

    Glitzky, A., Hünlich, R.: Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures. Appl. Anal. 66(3–4), 205–226 (1997)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Glitzky, A., Hünlich, R.: Electro-reaction-diffusion systems including cluster reactions of higher order. Math. Nachr. 216, 95–118 (2000)

    MATH  MathSciNet  Article  Google Scholar 

  31. 31.

    Gröger, K.: Free energy estimates and asymptotic behaviour of reaction-diffusion processes. Preprint 20, Institut für Angewandte Analysis und Stochastik im Forschungsverbund Berlin e.V. (1992)

  32. 32.

    Hall, R.N.: Electron-hole recombination in germanium. Phys. Rev. 87, 387–387 (1952)

    Article  Google Scholar 

  33. 33.

    Kufner, A., John, O., John, O., Fučík, S.: Function spaces. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics, Analysis. Noordhoff International Publishing, Leyden (1977)

  34. 34.

    Mack, C.: Fundamental Principles of Optical Lithography : The Science of Microfabrication. Wiley, Chichester (2007)

    Book  Google Scholar 

  35. 35.

    Matiut, D., Erdmann, A., Tollkuehn, B., Semmler, A.: New models for the simulation of post-exposure bake of chemically amplified resists, pp. 1132–1142 (2003)

  36. 36.

    Matthies, G., Tobiska, L.: Mass conservation of finite element methods for coupled flow-transport problems. Int. J. Comput. Sci. Math. 1(2–4), 293–307 (2007)

    MATH  MathSciNet  Article  Google Scholar 

  37. 37.

    Morgan, J.: Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20(5), 1128–1144 (1989)

    MATH  MathSciNet  Article  Google Scholar 

  38. 38.

    Moser, J.: A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13(3), 457–468 (1960)

    MATH  Article  Google Scholar 

  39. 39.

    Muirhead, R.F.: Some methods applicable to identities and inequalities of symmetric algebraic functions of \(n\) letters. Edinb. M. S. Proc. 21, 143–157 (1903)

    MATH  Google Scholar 

  40. 40.

    Nirenberg, L.: An extended interpolation inequality. Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat. III Ser. 20, 733–737 (1966)

    MATH  MathSciNet  Google Scholar 

  41. 41.

    Pierre, M.: Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78(2), 417–455 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  42. 42.

    Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with PARDISO. In: Computational Science-ICCS 2002, Part II (Amsterdam), Lecture Notes in Computer Science, vol. 2330, pp. 355–363. Springer, Berlin (2002)

  43. 43.

    Schenk, O., Gärtner, K.: On fast factorization pivoting methods for sparse symmetric indefinite systems. ETNA Electron. Trans. Numer. Anal. 23, 158–179 (2006)

    MATH  Google Scholar 

  44. 44.

    Shockley, W., Read, W.T.: Statistics of the recombinations of holes and electrons. Phys. Rev. 87, 835–842 (1952)

    MATH  Article  Google Scholar 

  45. 45.

    Zeidler, E., Boron, L.: Nonlinear functional analysis and its applications: II/B Nonlinear monotone operators. No. Bd. 2 in Nonlinear Functional Analysis and Its Applications. Springer, New York (1990)

Download references

Acknowledgments

A. Fiebach was supported by the European Commission within the Seventh Framework Programme \((FP7)\) MD3 “Material Development for Double Exposure and Double Patterning”. A. Linke and A. Glitzky are partially supported by the DFG Research Center Matheon Mathematics for key technologies. The authors would like to thank Klaus Gärtner, Jens Griepentrog, and Hang Si for their assistance and for valuable discussions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to André Fiebach.

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. 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. 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. 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. 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. 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. 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

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

\(\square \)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fiebach, A., Glitzky, A. & Linke, A. Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction–diffusion problems. Numer. Math. 128, 31–72 (2014). https://doi.org/10.1007/s00211-014-0604-6

Download citation

Mathematics Subject Classification (2000)

  • 35K57
  • 65M08
  • 65M22
  • 80A30