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Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution
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 |
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.
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. Google Scholar |
[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. Google Scholar |
[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. |
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. |
[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. Google Scholar |
[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. Google Scholar |
[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. |
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