# American Institute of Mathematical Sciences

July  2008, 20(3): 543-576. doi: 10.3934/dcds.2008.20.543

## Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients

 1 Department of Mathematics and Statistics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada, Canada 2 State Key Lab. of Scientific & Engin. Computing, Institute of Comput. Math. & Science-Engineering Computing, Chinese Academy of Sciences , Beijing, 100080, China

Received  November 2006 Revised  September 2007 Published  December 2007

In this paper we discuss initial-boundary problems for second order parabolic equations with rapidly oscillating coefficients in a bounded convex domain. The asymptotic expansions of the solutions for problems with multiple spatial and temporal scales are presented in four different cases. Higher order corrector methods are constructed and associated explicit convergence rates obtained.
Citation: Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543
 [1] Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks & Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711 [2] Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure & Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016 [3] Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105 [4] Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263 [5] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [6] Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017 [7] Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020 [8] Fernando Casas, Cristina Chiralt. A Lie--Deprit perturbation algorithm for linear differential equations with periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 959-975. doi: 10.3934/dcds.2014.34.959 [9] Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 [10] Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633 [11] Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 [12] Galina Kurina, Vladimir Zadorozhniy. Mean periodic solutions of a inhomogeneous heat equation with random coefficients. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-9. doi: 10.3934/dcdss.2020087 [13] Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3299-3318. doi: 10.3934/dcdsb.2018321 [14] Grégoire Allaire, M. Vanninathan. Homogenization of the Schrödinger equation with a time oscillating potential. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 1-16. doi: 10.3934/dcdsb.2006.6.1 [15] Micol Amar. A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 537-556. doi: 10.3934/dcds.2000.6.537 [16] Nobuyuki Kato, Norio Kikuchi. Campanato-type boundary estimates for Rothe's scheme to parabolic partial differential systems with constant coefficients. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 737-760. doi: 10.3934/dcds.2007.19.737 [17] Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154 [18] Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489 [19] Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 [20] Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

2018 Impact Factor: 1.143