Heterogeneous multiscale finite element method with novel numerical integration schemes

In this paper we introduce two novel numerical integration schemes, within the framework of the heterogeneous multiscale method (HMM) when finite element method is used as the macroscopic solver, to resolve the elliptic problem with multiscale coefficient. For non-self-adjoint elliptic problems, optimal convergence rate is proved for the proposed methods, which naturally yields a new strategy for refining the macro-micro meshes and a criterion for determining the size of the microcell. Numerical results following this strategy show that the new methods significantly reduce the computational cost without loss of accuracy.

[1]  E Weinan,et al.  The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size , 2007, J. Comput. Phys..

[2]  Assyr Abdulle,et al.  Heterogeneous Multiscale FEM for Diffusion Problems on Rough Surfaces , 2005, Multiscale Model. Simul..

[3]  M. Vogelius,et al.  Gradient Estimates for Solutions to Divergence Form Elliptic Equations with Discontinuous Coefficients , 2000 .

[4]  Assyr Abdulle,et al.  The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs , 2009 .

[5]  Ivo Babuška,et al.  Damage analysis of fiber composites Part I: Statistical analysis on fiber scale , 1999 .

[6]  Pingwen Zhang,et al.  Analysis of the heterogeneous multiscale method for parabolic homogenization problems , 2007, Math. Comput..

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

[8]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[9]  Assyr Abdulle,et al.  Erratum to ''A short and versatile finite element multiscale code for homogenization problems" (Comput. Methods Appl. Mech. Engrg. 198 (2009) 2839-2859) , 2010 .

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

[11]  E Weinan,et al.  The Heterogeneous Multiscale Method Based on the Discontinuous Galerkin Method for Hyperbolic and Parabolic Problems , 2005, Multiscale Model. Simul..

[12]  Assyr Abdulle,et al.  Finite Element Heterogeneous Multiscale Methods with Near Optimal Computational Complexity , 2008, Multiscale Model. Simul..

[13]  E Weinan,et al.  The Heterognous Multiscale Methods , 2003 .

[14]  Fabio Nobile,et al.  Worst case scenario analysis for elliptic problems with uncertainty , 2005, Numerische Mathematik.

[15]  S. Torquato,et al.  Scale effects on the elastic behavior of periodic andhierarchical two-dimensional composites , 1999 .

[16]  Ivo Babuška,et al.  Assessment of the cost and accuracy of the generalized FEM , 2007 .

[17]  C. Schwab,et al.  Two-scale FEM for homogenization problems , 2002 .

[18]  Assyr Abdulle,et al.  A short and versatile finite element multiscale code for homogenization problems , 2009 .

[19]  E Weinan,et al.  The heterogeneous multiscale method* , 2012, Acta Numerica.

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

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

[22]  A. Stroud,et al.  Numerical integration over simplexes , 1956 .

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

[24]  Yoon-ha Lee,et al.  Uncertainty Quantification for Multiscale Simulations , 2002 .

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