The Heterogeneous Multiscale Method Based on the Discontinuous Galerkin Method for Hyperbolic and Parabolic Problems

In this paper we develop a discontinuous Galerkin (DG) method, within the framework of the heterogeneous multiscale method (HMM), for solving hyperbolic and parabolic multiscale problems. Hyperbolic scalar equations and systems, as well as parabolic scalar problems, are considered. Error estimates are given for the linear equations, and numerical results are provided for the linear and nonlinear problems to demonstrate the capability of the method.

[1]  E Weinan,et al.  The Heterogeneous Multi-Scale Method , 2002 .

[2]  B. Engquist,et al.  Wavelet-Based Numerical Homogenization with Applications , 2002 .

[3]  Thomas Y. Hou,et al.  Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , 1999, Math. Comput..

[4]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[5]  Assyr Abdulle,et al.  On A Priori Error Analysis of Fully Discrete Heterogeneous Multiscale FEM , 2005, Multiscale Model. Simul..

[6]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[7]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[8]  G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization , 1989 .

[9]  E Weinan,et al.  Finite difference heterogeneous multi-scale method for homogenization problems , 2003 .

[10]  Ivo Babuska,et al.  Generalized p-FEM in homogenization , 2000, Numerische Mathematik.

[11]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[12]  E. Weinan,et al.  Analysis of the heterogeneous multiscale method for elliptic homogenization problems , 2004 .

[13]  E. Weinan.  Homogenization of linear and nonlinear transport equations , 1992 .

[14]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[15]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[16]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[17]  P. Donato,et al.  An introduction to homogenization , 2000 .

[18]  B. Engquist,et al.  Wavelet-Based Numerical Homogenization , 1998 .

[19]  C. Schwab,et al.  Generalized FEM for Homogenization Problems , 2002 .

[20]  Gabriel Wittum,et al.  Homogenization and Multigrid , 2001, Computing.

[21]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[22]  Bernardo Cockburn,et al.  Discontinuous Galerkin Methods for Convection-Dominated Problems , 1999 .

[23]  Chi-Wang Shu,et al.  The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws , 1988, ESAIM: Mathematical Modelling and Numerical Analysis.

[24]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[25]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[26]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[27]  Thomas Y. Hou,et al.  Computation of oscillatory solutions to hyperbolic differential equations using particle methods , 1988 .

[28]  Xingye Yue,et al.  Numerical methods for multiscale elliptic problems , 2006, J. Comput. Phys..

[29]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[30]  Endre Süli,et al.  Enhanced accuracy by post-processing for finite element methods for hyperbolic equations , 2003, Math. Comput..