Skip to main content

2018 | OriginalPaper | Buchkapitel

4. Corner Layers and Turning Points for Convection-Diffusion Equations

verfasst von : Gung-Min Gie, Makram Hamouda, Chang-Yeol Jung, Roger M. Temam

Erschienen in: Singular Perturbations and Boundary Layers

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this chapter and in Chapter 5, we investigate the boundary layers of convection-diffusion equations in space dimension one or two, and discuss additional issues to further develop the analysis performed in the previous Chapters 1 and 2

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Fußnoten
1
It is worth noting that imposing the outflow boundary conditions for (4.152) will not be consistent.
 
Literatur
[BHK84]
Zurück zum Zitat Alan E. Berger, Hou De Han, and R. Bruce Kellogg. A priori estimates and analysis of a numerical method for a turning point problem. Math. Comp., 42(166):465–492, 1984.MathSciNetCrossRef Alan E. Berger, Hou De Han, and R. Bruce Kellogg. A priori estimates and analysis of a numerical method for a turning point problem. Math. Comp., 42(166):465–492, 1984.MathSciNetCrossRef
[Can84]
Zurück zum Zitat John Rozier Cannon. The one-dimensional heat equation, volume 23 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, 1984. With a foreword by Felix E. Browder. John Rozier Cannon. The one-dimensional heat equation, volume 23 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, 1984. With a foreword by Felix E. Browder.
[DES87a]
Zurück zum Zitat Albert J. DeSanti. Nonmonotone interior layer theory for some singularly perturbed quasilinear boundary value problems with turning points. SIAM J. Math. Anal., 18(2):321–331, 1987.MathSciNetCrossRef Albert J. DeSanti. Nonmonotone interior layer theory for some singularly perturbed quasilinear boundary value problems with turning points. SIAM J. Math. Anal., 18(2):321–331, 1987.MathSciNetCrossRef
[DES87b]
Zurück zum Zitat Albert J. DeSanti. Perturbed quasilinear Dirichlet problems with isolated turning points. Comm. Partial Differential Equations, 12(2):223–242, 1987.MathSciNetCrossRef Albert J. DeSanti. Perturbed quasilinear Dirichlet problems with isolated turning points. Comm. Partial Differential Equations, 12(2):223–242, 1987.MathSciNetCrossRef
[DGZ07]
Zurück zum Zitat Yihong Du, Zongming Guo, and Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete Contin. Dyn. Syst., 19(2):271–298, 2007.MathSciNetCrossRef Yihong Du, Zongming Guo, and Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete Contin. Dyn. Syst., 19(2):271–298, 2007.MathSciNetCrossRef
[DL13]
Zurück zum Zitat Zhuoran Du and Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete Contin. Dyn. Syst., 33(4):1407–1429, 2013.MathSciNetMATH Zhuoran Du and Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete Contin. Dyn. Syst., 33(4):1407–1429, 2013.MathSciNetMATH
[Gie14]
Zurück zum Zitat Gung-Min Gie. Asymptotic expansion of the Stokes solutions at small viscosity: the case of non-compatible initial data. Commun. Math. Sci., 12(2):383–400, 2014.MathSciNetCrossRef Gung-Min Gie. Asymptotic expansion of the Stokes solutions at small viscosity: the case of non-compatible initial data. Commun. Math. Sci., 12(2):383–400, 2014.MathSciNetCrossRef
[GHS15]
Zurück zum Zitat Gung-Min Gie, Makram Hamouda, and Abir Sboui. Asymptotic analysis of the Stokes equations in a square at small viscosity. Appl. Anal. 95 (2016), no. 12, 2683–2702.MathSciNetCrossRef Gung-Min Gie, Makram Hamouda, and Abir Sboui. Asymptotic analysis of the Stokes equations in a square at small viscosity. Appl. Anal. 95 (2016), no. 12, 2683–2702.MathSciNetCrossRef
[GHT10b]
Zurück zum Zitat Gung-Min Gie, Makram Hamouda, and Roger Temam. Boundary layers in smooth curvilinear domains: parabolic problems. Discrete Contin. Dyn. Syst., 26(4):1213–1240, 2010.MathSciNetMATH Gung-Min Gie, Makram Hamouda, and Roger Temam. Boundary layers in smooth curvilinear domains: parabolic problems. Discrete Contin. Dyn. Syst., 26(4):1213–1240, 2010.MathSciNetMATH
[GJT13]
Zurück zum Zitat Gung-Min Gie, Chang-Yeol Jung, and Roger Temam. Analysis of mixed elliptic and parabolic boundary layers with corners. Int. J. Differ. Equ., pages Art. ID 532987, 13, 2013. Gung-Min Gie, Chang-Yeol Jung, and Roger Temam. Analysis of mixed elliptic and parabolic boundary layers with corners. Int. J. Differ. Equ., pages Art. ID 532987, 13, 2013.
[GJT16]
Zurück zum Zitat Gung-Min Gie, Chang-Yeol Jung, and Roger Temam. Recent progresses in boundary layer theory. Discrete Contin. Dyn. Syst., 36(5):2521–2583, 2016.MathSciNetMATH Gung-Min Gie, Chang-Yeol Jung, and Roger Temam. Recent progresses in boundary layer theory. Discrete Contin. Dyn. Syst., 36(5):2521–2583, 2016.MathSciNetMATH
[GKLMN18]
Zurück zum Zitat Gung-Min Gie, James P. Kelliher, M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes. Vanishing viscosity limit of some symmetric flows. Preprint, 2018. Gung-Min Gie, James P. Kelliher, M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes. Vanishing viscosity limit of some symmetric flows. Preprint, 2018.
[Gri92]
Zurück zum Zitat P. Grisvard. Singularities in boundary value problems, volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris; Springer-Verlag, Berlin, 1992. P. Grisvard. Singularities in boundary value problems, volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris; Springer-Verlag, Berlin, 1992.
[Gri11]
Zurück zum Zitat P. Grisvard. Elliptic problems in nonsmooth domains, volume 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Reprint of the 1985 original, With a foreword by Susanne C. Brenner. P. Grisvard. Elliptic problems in nonsmooth domains, volume 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Reprint of the 1985 original, With a foreword by Susanne C. Brenner.
[JN13]
Zurück zum Zitat Chang-Yeol Jung and Thien Binh Nguyen. Semi-analytical numerical methods for convection-dominated problems with turning points. Int. J. Numer. Anal. Model., 10(2):314–332, 2013. Chang-Yeol Jung and Thien Binh Nguyen. Semi-analytical numerical methods for convection-dominated problems with turning points. Int. J. Numer. Anal. Model., 10(2):314–332, 2013.
[JPT16]
Zurück zum Zitat Chang-Yeol Jung, Eunhee Park, and Roger Temam. Boundary layer analysis of nonlinear reaction-diffusion equations in a smooth domain. Adv. Nonlinear Anal., 6(3):277–300, 2017MathSciNetMATH Chang-Yeol Jung, Eunhee Park, and Roger Temam. Boundary layer analysis of nonlinear reaction-diffusion equations in a smooth domain. Adv. Nonlinear Anal., 6(3):277–300, 2017MathSciNetMATH
[JPT17]
Zurück zum Zitat Chang-Yeol Jung, Eunhee Park, and Roger Temam. Boundary layer analysis of nonlinear reaction-diffusion equations in a polygonal domain. Nonlinear Anal., 148:161–202, 2017.MathSciNetCrossRef Chang-Yeol Jung, Eunhee Park, and Roger Temam. Boundary layer analysis of nonlinear reaction-diffusion equations in a polygonal domain. Nonlinear Anal., 148:161–202, 2017.MathSciNetCrossRef
[JT05]
Zurück zum Zitat Chang-Yeol Jung and Roger Temam. Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers. Int. J. Numer. Anal. Model., 2(4):367–408, 2005.MathSciNetMATH Chang-Yeol Jung and Roger Temam. Numerical approximation of two-dimensional convection-diffusion equations with multiple boundary layers. Int. J. Numer. Anal. Model., 2(4):367–408, 2005.MathSciNetMATH
[JT07]
Zurück zum Zitat Chang-Yeol Jung and Roger Temam. Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point. J. Math. Phys., 48(6):065301, 27, 2007. Chang-Yeol Jung and Roger Temam. Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point. J. Math. Phys., 48(6):065301, 27, 2007.
[JT09b]
Zurück zum Zitat Chang-Yeol Jung and Roger Temam. Interaction of boundary layers and corner singularities. Discrete Contin. Dyn. Syst., 23(1–2):315–339, 2009.MathSciNetMATH Chang-Yeol Jung and Roger Temam. Interaction of boundary layers and corner singularities. Discrete Contin. Dyn. Syst., 23(1–2):315–339, 2009.MathSciNetMATH
[JT14b]
Zurück zum Zitat Chang-Yeol Jung and Roger Temam. Singularly perturbed problems with a turning point: the non-compatible case. Anal. Appl. (Singap.), 12(3):293–321, 2014.MathSciNetCrossRef Chang-Yeol Jung and Roger Temam. Singularly perturbed problems with a turning point: the non-compatible case. Anal. Appl. (Singap.), 12(3):293–321, 2014.MathSciNetCrossRef
[KK10]
Zurück zum Zitat R. Bruce Kellogg and Natalia Kopteva. A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain. (English summary) J. Differential Equations, 248 (2010), no. 1, 184–208. R. Bruce Kellogg and Natalia Kopteva. A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain. (English summary) J. Differential Equations, 248 (2010), no. 1, 184–208.
[KS05]
Zurück zum Zitat R. Bruce Kellogg and Martin Stynes. Corner singularities and boundary layers in a simple convection-diffusion problem. J. Differential Equations, 213(1):81–120, 2005.MathSciNetCrossRef R. Bruce Kellogg and Martin Stynes. Corner singularities and boundary layers in a simple convection-diffusion problem. J. Differential Equations, 213(1):81–120, 2005.MathSciNetCrossRef
[LMN08]
Zurück zum Zitat M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes. Vanishing viscosity limit for incompressible flow inside a rotating circle. Phys. D, 237(10–12):1324–1333, 2008.MathSciNetCrossRef M. C. Lopes Filho, A. L. Mazzucato, and H. J. Nussenzveig Lopes. Vanishing viscosity limit for incompressible flow inside a rotating circle. Phys. D, 237(10–12):1324–1333, 2008.MathSciNetCrossRef
[LMNT08]
Zurück zum Zitat M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, and Michael Taylor. Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows. Bull. Braz. Math. Soc. (N.S.), 39(4):471–513, 2008.MathSciNetCrossRef M. C. Lopes Filho, A. L. Mazzucato, H. J. Nussenzveig Lopes, and Michael Taylor. Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows. Bull. Braz. Math. Soc. (N.S.), 39(4):471–513, 2008.MathSciNetCrossRef
[NN12]
Zurück zum Zitat N. N. Nefedov and A. G. Nikitin. The initial boundary value problem for a nonlocal singularly perturbed reaction-diffusion equation. (English summary) Comput. Math. Math. Phys. 52 (2012), no. 6, 926–931; translated from Zh. Vychisl. Mat. Mat. Fiz. 52 (2012), no. 6, 1042–1047 (Russian. Russian summary).MathSciNetCrossRef N. N. Nefedov and A. G. Nikitin. The initial boundary value problem for a nonlocal singularly perturbed reaction-diffusion equation. (English summary) Comput. Math. Math. Phys. 52 (2012), no. 6, 926–931; translated from Zh. Vychisl. Mat. Mat. Fiz. 52 (2012), no. 6, 1042–1047 (Russian. Russian summary).MathSciNetCrossRef
[OMa70]
Zurück zum Zitat R. E. O’Malley, Jr. On boundary value problems for a singularly perturbed differential equation with a turning point. SIAM J. Math. Anal., 1:479–490, 1970.MathSciNetCrossRef R. E. O’Malley, Jr. On boundary value problems for a singularly perturbed differential equation with a turning point. SIAM J. Math. Anal., 1:479–490, 1970.MathSciNetCrossRef
[OQ15]
Zurück zum Zitat O’Riordan, E.; Quinn, J. A linearised singularly perturbed convection-diffusion problem with an interior layer. Appl. Numer. Math., 98 (2015), 1–17.MathSciNetCrossRef O’Riordan, E.; Quinn, J. A linearised singularly perturbed convection-diffusion problem with an interior layer. Appl. Numer. Math., 98 (2015), 1–17.MathSciNetCrossRef
[SK87]
Zurück zum Zitat Shagi-Di Shih and R. Bruce Kellogg. Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal., 18(5):1467–1511, 1987.MathSciNetCrossRef Shagi-Di Shih and R. Bruce Kellogg. Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal., 18(5):1467–1511, 1987.MathSciNetCrossRef
[SS94]
Zurück zum Zitat Guang Fu Sun and Martin Stynes. Finite element methods on piecewise equidistant meshes for interior turning point problems. Numer. Algorithms, 8(1):111–129, 1994. Guang Fu Sun and Martin Stynes. Finite element methods on piecewise equidistant meshes for interior turning point problems. Numer. Algorithms, 8(1):111–129, 1994.
[WAS85]
Zurück zum Zitat Wolfgang Wasow. Linear turning point theory, volume 54 of Applied Mathematical Sciences. Springer-Verlag, New York, 1985.CrossRef Wolfgang Wasow. Linear turning point theory, volume 54 of Applied Mathematical Sciences. Springer-Verlag, New York, 1985.CrossRef
[WY02a]
[WY02b]
Zurück zum Zitat R. Wong and Heping Yang. On an internal boundary layer problem. J. Comput. Appl. Math., 144(1–2):301–323, 2002.MathSciNetCrossRef R. Wong and Heping Yang. On an internal boundary layer problem. J. Comput. Appl. Math., 144(1–2):301–323, 2002.MathSciNetCrossRef
[WY03]
Zurück zum Zitat R. Wong and Heping Yang. On the Ackerberg-O’Malley resonance. Stud. Appl. Math., 110(2):157–179, 2003.MathSciNetCrossRef R. Wong and Heping Yang. On the Ackerberg-O’Malley resonance. Stud. Appl. Math., 110(2):157–179, 2003.MathSciNetCrossRef
[WZ06]
Zurück zum Zitat R. Wong and Y. Zhao. A singularly perturbed boundary-value problem arising in phase transitions. European J. Appl. Math., 17(6):705–733, 2006.MathSciNetCrossRef R. Wong and Y. Zhao. A singularly perturbed boundary-value problem arising in phase transitions. European J. Appl. Math., 17(6):705–733, 2006.MathSciNetCrossRef
Metadaten
Titel
Corner Layers and Turning Points for Convection-Diffusion Equations
verfasst von
Gung-Min Gie
Makram Hamouda
Chang-Yeol Jung
Roger M. Temam
Copyright-Jahr
2018
DOI
https://doi.org/10.1007/978-3-030-00638-9_4