November  2007, 8(4): 833-859. doi: 10.3934/dcdsb.2007.8.833

Multiscale methods for parabolic equations with continuum spatial scales

1. 

Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States

2. 

Department of Mathematics & ISC, Texas A&M University, 3404 TAMU, College Station, TX 77843-3404, United States

3. 

Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, United States

Received  March 2007 Revised  June 2007 Published  August 2007

In this paper, we consider multiscale approaches for solving parabolic equations with heterogeneous coefficients. Our interest stems from porous media applications and we assume that there is no scale separation with respect to spatial variables. To compute the solution of these multiscale problems on a coarse grid, we define global fields such that the solution smoothly depends on these fields. We present various finite element discretization techniques and provide analyses of these methods. A few representative numerical examples are presented using heterogeneous fields with strong non-local features. These numerical results demonstrate that the solution can be captured more accurately on the coarse grid when some type of limited global information is used.
Citation: Lijian Jiang, Yalchin Efendiev, Victor Ginting. Multiscale methods for parabolic equations with continuum spatial scales. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 833-859. doi: 10.3934/dcdsb.2007.8.833
[1]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449

[2]

Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543

[3]

Sandra Lucente. Global existence for equivalent nonlinear special scale invariant damped wave equations. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021159

[4]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074

[5]

Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11

[6]

Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete and Continuous Dynamical Systems, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53

[7]

Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4637-4676. doi: 10.3934/dcds.2017200

[8]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034

[9]

Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure and Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923

[10]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1

[11]

Boling Guo, Guoli Zhou. Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4305-4327. doi: 10.3934/dcdsb.2018160

[12]

Dung Le. Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 877-893. doi: 10.3934/dcdsb.2017044

[13]

Yoonsang Lee, Bjorn Engquist. Variable step size multiscale methods for stiff and highly oscillatory dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1079-1097. doi: 10.3934/dcds.2014.34.1079

[14]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks and Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[15]

Fabio Punzo. Global solutions of semilinear parabolic equations with drift term on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022030

[16]

Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667

[17]

Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281

[18]

Stephan Didas, Joachim Weickert. Integrodifferential equations for continuous multiscale wavelet shrinkage. Inverse Problems and Imaging, 2007, 1 (1) : 47-62. doi: 10.3934/ipi.2007.1.47

[19]

Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631

[20]

Yang Yu. Introduction: Special issue on computational intelligence methods for big data and information analytics. Big Data & Information Analytics, 2017, 2 (1) : i-ii. doi: 10.3934/bdia.201701i

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (121)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]