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

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• 作者：André Fiebach (1)
  Annegret Glitzky (2)
  Alexander Linke (2)
  
• 关键词：35K57 ; 65M08 ; 65M22 ; 80A30
• 刊名：Numerische Mathematik
• 出版年：2014
• 出版时间：September 2014
• 年：2014
• 卷：128
• 期：1
• 页码：31-72
• 全文大小：1,176 KB
• 参考文献：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. Bothe, D., Pierre, M.: Quasi-steady-state approximation for a reaction-diffusion system with fast intermediate. J. Math. Anal. Appl. 368(1), 120-32 (2010) CrossRef
  3. Brown, A.J.: Enzym action. J. Chem. Soc. 81, 373-86 (1902) CrossRef
  4. Carberry, J.: Chemical and catalytic reaction engineering. Dover Books on Chemistry Series. Dover Publications, Dover (2001)
  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), 660-80 (2000) CrossRef
  6. Desvillettes, L., Fellner, K.: Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations. J. Math. Anal. Appl. 319(1), 157-76 (2006) CrossRef
  7. Desvillettes, L., Fellner, K., Pierre, M., Vovelle, J.: Global existence for quadratic systems of reaction-diffusion. Adv Nonlinear Stud 7(3), 491-11 (2007)
  8. Deuflhard, P., Bornemann, F.: Gew?hnliche Differentialgleichungen. No. Bd. 2 in Numerische Mathematik. de Gruyter, Germany (2008)
  9. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
  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. Eymard, R., Fuhrmann, J., G?rtner, K.: A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems. Numer. Math. 102(3), 463-95 (2006) CrossRef
  12. Eymard, R., Gallou?t, T., Herbin, R.: The finite volume method. In: Ciarlet, P., Lions, J.L. (eds.) Handbook of Numerical Analysis, pp. 713-020. North Holland, Amsterdam (2000)
  13. Eymard, R., Gallou?t, 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), 1009-043 (2010). doi:10.1093/imanum/drn084
  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), 631-54 (2010). doi:10.1007/s11401-010-0604-5
  15. Fiebach, A.: A dissipative finite volume scheme for reaction-diffusion systems in heterogeneous materials. Ph.D. thesis, Freie Universit?t Berlin (Submitted December 2013)
  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. 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-), 792-95 (2009) CrossRef
  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), 530-53 (2011) CrossRef
  19. Gajewski, H., Gr?ger, K.: Reaction–diffusion processes of electrically charged species. Math. Nachr. 177(1), 109-30 (1996) CrossRef
  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. 151-88. Birkh?user, Basel (2005)
  21. G?rtner, K.: Existence of bounded discrete steady-state solutions of the van Roosbroeck system on boundary conforming Delaunay grids. SIAM J. Sci. Comput. 31(2), 1347-362 (2009) CrossRef
  22. Giovangigli, V.: Multicomponent flow modeling. Modeling and Simulation in Science, Engineering & Technology. Birkh?user, Boston (1999)
  23. Glitzky, A.: Exponential decay of the free energy for discretized electro-reaction-diffusion systems. Nonlinearity 21(9), 1989-009 (2008) CrossRef
  24. Glitzky, A.: Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction-diffusion systems. Math. Nachr. 284(17-8), 2159-174 (2011) CrossRef
  25. Glitzky, A., G?rtner, K.: Energy estimates for continuous and discretized electro-reaction-diffusion systems. Nonlinear Anal. Theory Methods Appl. Ser. A 70(2), 788-05 (2009) CrossRef
  26. Glitzky, A., Griepentrog, J.A.: Discrete Sobolev–Poincaré inequalities for Voronoi finite volume approximations. SIAM J. Numer. Anal. 48(1), 372-91 (2010) CrossRef
  27. Glitzky, A., Gr?ger, K., Hünlich, R.: Free energy and dissipation rate for reaction diffusion processes of electrically charged species. Appl. Anal. 60(3-), 201-17 (1996) CrossRef
  28. Glitzky, A., Hünlich, R.: Energetic estimates and asymptotics for electro-reaction-diffusion systems. Z. Angew. Math. Mech. 77(11), 823-32 (1997) CrossRef
  29. Glitzky, A., Hünlich, R.: Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures. Appl. Anal. 66(3-), 205-26 (1997) CrossRef
  30. Glitzky, A., Hünlich, R.: Electro-reaction-diffusion systems including cluster reactions of higher order. Math. Nachr. 216, 95-18 (2000) CrossRef
  31. Gr?ger, 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. Hall, R.N.: Electron-hole recombination in germanium. Phys. Rev. 87, 387-87 (1952) CrossRef
  33. Kufner, A., John, O., John, O., Fu?ík, S.: Function spaces. Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics, Analysis. Noordhoff International Publishing, Leyden (1977)
  34. Mack, C.: Fundamental Principles of Optical Lithography : The Science of Microfabrication. Wiley, Chichester (2007) CrossRef
  35. Matiut, D., Erdmann, A., Tollkuehn, B., Semmler, A.: New models for the simulation of post-exposure bake of chemically amplified resists, pp. 1132-142 (2003)
  36. Matthies, G., Tobiska, L.: Mass conservation of finite element methods for coupled flow-transport problems. Int. J. Comput. Sci. Math. 1(2-), 293-07 (2007) CrossRef
  37. Morgan, J.: Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20(5), 1128-144 (1989) CrossRef
  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), 457-68 (1960) CrossRef
  39. Muirhead, R.F.: Some methods applicable to identities and inequalities of symmetric algebraic functions of \(n\) letters. Edinb. M. S. Proc. 21, 143-57 (1903)
  40. Nirenberg, L.: An extended interpolation inequality. Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat. III Ser. 20, 733-37 (1966)
  41. Pierre, M.: Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78(2), 417-55 (2010) CrossRef
  42. Schenk, O., G?rtner, 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. 355-63. Springer, Berlin (2002)
  43. Schenk, O., G?rtner, K.: On fast factorization pivoting methods for sparse symmetric indefinite systems. ETNA Electron. Trans. Numer. Anal. 23, 158-79 (2006)
  44. Shockley, W., Read, W.T.: Statistics of the recombinations of holes and electrons. Phys. Rev. 87, 835-42 (1952) CrossRef
  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)
  
• 作者单位：André Fiebach (1)
  Annegret Glitzky (2)
  Alexander Linke (2)
  
  1. Physikalisch-Technische Bundesanstalt (PTB), Abbestra?e 2-12, 10587?, Berlin, Germany
  2. Weierstrass Institute, Leibniz Institute in Forschungsverbund Berlin e.V. (WIAS), Mohrenstr. 39, 10117?, Berlin, Germany
  
• ISSN：0945-3245

文摘

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.