January  2020, 25(1): 335-399. doi: 10.3934/dcdsb.2019186

Multi-scale analysis for highly anisotropic parabolic problems

Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille UMR 7373, Château Gombert 39 rue F. Joliot Curie, Marseille, 13453, FRANCE

* Corresponding author: Mihaï Bostan

Received  February 2018 Revised  February 2019 Published  September 2019

We focus on the asymptotic behavior of strongly anisotropic parabolic problems. We concentrate on heat equations, whose diffusion matrix fields have disparate eigenvalues. We establish strong convergence results toward a profile. Under suitable smoothness hypotheses, by introducing an appropriate corrector term, we estimate the convergence rate. The arguments rely on two-scale analysis, based on average operators with respect to unitary groups.

Citation: Thomas Blanc, Mihaï Bostan. Multi-scale analysis for highly anisotropic parabolic problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 335-399. doi: 10.3934/dcdsb.2019186
References:
[1]

L. Agelas and R. Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Math. Acad. Sci. Paris, 346 (2008), 1007-1012. doi: 10.1016/j.crma.2008.07.015. Google Scholar

[2]

D. S. BalsanaD. A. Tilley and C. J. Howk, Simulating anisotropic thermal conduction in supernova remnants-Ⅰ., Numerical methods, Monthly Notices of Royal Astronomical Society, 386 (2008), 627-641. Google Scholar

[3]

T. BlancM. Bostan and F. Boyer, Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4637-4676. doi: 10.3934/dcds.2017200. Google Scholar

[4]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010. Google Scholar

[5]

M. Bostan, Strongly anisotropic diffusion problems; asymptotic analysis, J. Differential Equations, 256 (2016), 1043-1092. doi: 10.1016/j.jde.2013.10.008. Google Scholar

[6]

M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188. doi: 10.1137/15M1033034. Google Scholar

[7]

S. I. Braginskii, Transport Processes in a Plasma, M.A. Leontovich, Reviews of Plasma Physics, Consultants Bureau, New York, 1965.Google Scholar

[8]

C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, 2009. doi: 10.1007/978-0-387-09819-7. Google Scholar

[9]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique Pour Les Sciences et Les Techniques, vol. 8, Masson, 1988. Google Scholar

[10]

P. DegondF. Deluzet and C. Negulescu, An asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul., 8 (2009/10), 645-666. doi: 10.1137/090754200. Google Scholar

[11]

R. EymardT. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043. doi: 10.1093/imanum/drn084. Google Scholar

[12]

F. FilbetC. Negulescu and C. Yang, Numerical study of a nonlinear heat equation for plasma physics, Int. J. Comput. Math., 89 (2012), 1060-1082. doi: 10.1080/00207160.2012.679732. Google Scholar

[13]

E. FreireA. Gasull and A. Guillamon, A characterization of isochronous centers in terms of symmetries, Rev. Mat. Iberoamericana, 20 (2004), 205-222. doi: 10.4171/RMI/386. Google Scholar

[14]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. Ⅰ, Springer Berlin Heidelberg, 1972. Google Scholar

[15]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639. doi: 10.1109/34.56205. Google Scholar

[16]

J. Quah and D. Margetis, Anisotropic diffusion in continuum relaxation of stepped crystal surfaces, J. Phys. A, 41 (2008), 235004, 18pp. doi: 10.1088/1751-8113/41/23/235004. Google Scholar

[17]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. Ⅰ, Functional Analysis, Academic Press, 1980. Google Scholar

[18]

M. Sabatini, Characterizing isochronous centers by Lie brackets, Differential Equations Dyn. Syst., 5 (1997), 91-99. Google Scholar

[19]

P. Sharma and G. W. Hammett, A fast semi-implicit method for anisotropic diffusion, J. Comput. Phys., 230 (2011), 4899-4909. doi: 10.1016/j.jcp.2011.03.009. Google Scholar

[20]

P. Sharma and G. W. Hammett, Preserving monotonicity in anisotropic diffusion, J. Comput. Phys., 227 (2007), 123-142. doi: 10.1016/j.jcp.2007.07.026. Google Scholar

[21]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart 1998. Google Scholar

show all references

References:
[1]

L. Agelas and R. Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes, C. R. Math. Acad. Sci. Paris, 346 (2008), 1007-1012. doi: 10.1016/j.crma.2008.07.015. Google Scholar

[2]

D. S. BalsanaD. A. Tilley and C. J. Howk, Simulating anisotropic thermal conduction in supernova remnants-Ⅰ., Numerical methods, Monthly Notices of Royal Astronomical Society, 386 (2008), 627-641. Google Scholar

[3]

T. BlancM. Bostan and F. Boyer, Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 4637-4676. doi: 10.3934/dcds.2017200. Google Scholar

[4]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010. Google Scholar

[5]

M. Bostan, Strongly anisotropic diffusion problems; asymptotic analysis, J. Differential Equations, 256 (2016), 1043-1092. doi: 10.1016/j.jde.2013.10.008. Google Scholar

[6]

M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188. doi: 10.1137/15M1033034. Google Scholar

[7]

S. I. Braginskii, Transport Processes in a Plasma, M.A. Leontovich, Reviews of Plasma Physics, Consultants Bureau, New York, 1965.Google Scholar

[8]

C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, 2009. doi: 10.1007/978-0-387-09819-7. Google Scholar

[9]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique Pour Les Sciences et Les Techniques, vol. 8, Masson, 1988. Google Scholar

[10]

P. DegondF. Deluzet and C. Negulescu, An asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul., 8 (2009/10), 645-666. doi: 10.1137/090754200. Google Scholar

[11]

R. EymardT. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), 1009-1043. doi: 10.1093/imanum/drn084. Google Scholar

[12]

F. FilbetC. Negulescu and C. Yang, Numerical study of a nonlinear heat equation for plasma physics, Int. J. Comput. Math., 89 (2012), 1060-1082. doi: 10.1080/00207160.2012.679732. Google Scholar

[13]

E. FreireA. Gasull and A. Guillamon, A characterization of isochronous centers in terms of symmetries, Rev. Mat. Iberoamericana, 20 (2004), 205-222. doi: 10.4171/RMI/386. Google Scholar

[14]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. Ⅰ, Springer Berlin Heidelberg, 1972. Google Scholar

[15]

P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639. doi: 10.1109/34.56205. Google Scholar

[16]

J. Quah and D. Margetis, Anisotropic diffusion in continuum relaxation of stepped crystal surfaces, J. Phys. A, 41 (2008), 235004, 18pp. doi: 10.1088/1751-8113/41/23/235004. Google Scholar

[17]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. Ⅰ, Functional Analysis, Academic Press, 1980. Google Scholar

[18]

M. Sabatini, Characterizing isochronous centers by Lie brackets, Differential Equations Dyn. Syst., 5 (1997), 91-99. Google Scholar

[19]

P. Sharma and G. W. Hammett, A fast semi-implicit method for anisotropic diffusion, J. Comput. Phys., 230 (2011), 4899-4909. doi: 10.1016/j.jcp.2011.03.009. Google Scholar

[20]

P. Sharma and G. W. Hammett, Preserving monotonicity in anisotropic diffusion, J. Comput. Phys., 227 (2007), 123-142. doi: 10.1016/j.jcp.2007.07.026. Google Scholar

[21]

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart 1998. Google Scholar

[1]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753

[2]

Stephen W. Taylor. Locally smooth unitary groups and applications to boundary control of PDEs. Evolution Equations & Control Theory, 2013, 2 (4) : 733-740. doi: 10.3934/eect.2013.2.733

[3]

Dag Lukkassen, Annette Meidell, Peter Wall. Multiscale homogenization of monotone operators. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 711-727. doi: 10.3934/dcds.2008.22.711

[4]

Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181

[5]

Aihua Fan, Lingmin Liao, Jacques Peyrière. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1103-1128. doi: 10.3934/dcds.2008.21.1103

[6]

Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks & Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181

[7]

Luciana A. Alves, Luiz A. B. San Martin. Multiplicative ergodic theorem on flag bundles of semi-simple Lie groups. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1247-1273. doi: 10.3934/dcds.2013.33.1247

[8]

Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677

[9]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086

[10]

Luis Caffarelli, Antoine Mellet. Random homogenization of fractional obstacle problems. Networks & Heterogeneous Media, 2008, 3 (3) : 523-554. doi: 10.3934/nhm.2008.3.523

[11]

Wenjia Jing, Panagiotis E. Souganidis, Hung V. Tran. Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 915-939. doi: 10.3934/dcdss.2018055

[12]

Alberto Boscaggin, Maurizio Garrione. Positive solutions to indefinite Neumann problems when the weight has positive average. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5231-5244. doi: 10.3934/dcds.2016028

[13]

Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787

[14]

Andrea Braides, Valeria Chiadò Piat. Non convex homogenization problems for singular structures. Networks & Heterogeneous Media, 2008, 3 (3) : 489-508. doi: 10.3934/nhm.2008.3.489

[15]

Sara Monsurrò, Carmen Perugia. Homogenization and exact controllability for problems with imperfect interface. Networks & Heterogeneous Media, 2019, 14 (2) : 411-444. doi: 10.3934/nhm.2019017

[16]

Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695

[17]

Fengping Yao. Optimal regularity for parabolic Schrödinger operators. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1407-1414. doi: 10.3934/cpaa.2013.12.1407

[18]

Barbara Kaltenbacher, William Rundell. Regularization of a backwards parabolic equation by fractional operators. Inverse Problems & Imaging, 2019, 13 (2) : 401-430. doi: 10.3934/ipi.2019020

[19]

Velimir Jurdjevic. Affine-quadratic problems on Lie groups. Mathematical Control & Related Fields, 2013, 3 (3) : 347-374. doi: 10.3934/mcrf.2013.3.347

[20]

Richard Miles, Michael Björklund. Entropy range problems and actions of locally normal groups. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 981-989. doi: 10.3934/dcds.2009.25.981

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (8)
  • HTML views (70)
  • Cited by (0)

Other articles
by authors

[Back to Top]