Multi-scale analysis for highly anisotropic parabolic problems

Thomas Blanc; Mihaï Bostan

doi:10.3934/dcdsb.2019186

Article Contents

2020, Volume 25, Issue 1: 335-399. Doi: 10.3934/dcdsb.2019186

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Multi-scale analysis for highly anisotropic parabolic problems

Thomas Blanc and
Mihaï Bostan^,

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
^* Corresponding author: Mihaï Bostan

Received: January 31, 2018

Revised: January 31, 2019

Early access: September 2019

Published: January 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K05; Secondary: 37A10.

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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.
[2]	D. S. Balsana, D. 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.
[3]	T. Blanc, M. 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.
[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.
[5]	M. Bostan, Strongly anisotropic diffusion problems; asymptotic analysis, J. Differential Equations, 256 (2016), 1043-1092. doi: 10.1016/j.jde.2013.10.008.
[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.
[7]	S. I. Braginskii, Transport Processes in a Plasma, M.A. Leontovich, Reviews of Plasma Physics, Consultants Bureau, New York, 1965.
[8]	C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, 2009. doi: 10.1007/978-0-387-09819-7.
[9]	R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique Pour Les Sciences et Les Techniques, vol. 8, Masson, 1988.
[10]	P. Degond, F. 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.
[11]	R. Eymard, T. 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.
[12]	F. Filbet, C. 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.
[13]	E. Freire, A. 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.
[14]	J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. Ⅰ, Springer Berlin Heidelberg, 1972.
[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.
[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.
[17]	M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. Ⅰ, Functional Analysis, Academic Press, 1980.
[18]	M. Sabatini, Characterizing isochronous centers by Lie brackets, Differential Equations Dyn. Syst., 5 (1997), 91-99.
[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.
[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.
[21]	J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart 1998.