An Efficient High Order Heterogeneous Multiscale Method for Elliptic Problems

We propose an efficient heterogeneous multiscale finite element method based on a local least-squares reconstruction of the effective matrix using the data retrieved from the solution of cell problems posed on the vertices of the triangulation. The method achieves high order accuracy for high order macroscopic solver with essentially the same cost as the linear macroscopic solver. Optimal error bounds are proved for the elliptic problem. Numerical results demonstrate that the new method significantly reduces the cost without loss of accuracy.

[1]  Zhimin Zhang,et al.  A New Finite Element Gradient Recovery Method: Superconvergence Property , 2005, SIAM J. Sci. Comput..

[2]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[3]  F. John.  Partial differential equations , 1967 .

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

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

[6]  Peter Wriggers,et al.  An Introduction to Computational Micromechanics , 2004 .

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

[8]  I. Doležel,et al.  Higher-Order Finite Element Methods , 2003 .

[9]  T. J. Rivlin,et al.  The growth of polynomials bounded at equally spaced points , 1992 .

[10]  Marcus J. Grote,et al.  Finite Element Heterogeneous Multiscale Method for the Wave Equation , 2011, Multiscale Model. Simul..

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

[12]  E. Rakhmanov,et al.  Bounds for polynomials with a unit discrete norm , 2007 .

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

[14]  Antoine Gloria,et al.  Reduction of the resonance error---Part 1: Approximation of homogenized coefficients , 2011 .

[15]  V. Venkatakrishnan.  Convergence to steady state solutions of the Euler equations on unstructured grids with limiters , 1995 .

[16]  V. Thomée,et al.  The Stability in- L and W^ of the L2-Projection onto Finite Element Function Spaces , 2010 .

[17]  Philip Rabinowitz,et al.  Perfectly symmetric two-dimensional integration formulas with minimal numbers of points , 1969 .

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

[19]  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..

[20]  H. Hadwiger,et al.  Einige Relationen im Dreieck , 1937 .

[21]  Endre Süli,et al.  Residual-free bubbles for advection-diffusion problems: the general error analysis , 2000, Numerische Mathematik.

[22]  Björn Engquist,et al.  Multiscale methods for the wave equation , 2007 .

[23]  Grégoire Allaire,et al.  A Multiscale Finite Element Method for Numerical Homogenization , 2005, Multiscale Model. Simul..

[24]  V. A. Kondrat'ev,et al.  Boundary problems for elliptic equations in domains with conical or angular points , 1967 .

[25]  Pingbing Ming,et al.  Heterogeneous multiscale finite element method with novel numerical integration schemes , 2010 .

[26]  Antoine Gloria,et al.  An Analytical Framework for the Numerical Homogenization of Monotone Elliptic Operators and Quasiconvex Energies , 2006, Multiscale Model. Simul..

[27]  Christian Huet,et al.  Order relationships for boundary conditions effect in heterogeneous bodies smaller than the representative volume , 1994 .

[28]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[29]  Lothar Reichel.  On polynomial approximation in the uniform norm by the discrete least squares method , 1986 .

[30]  M. Powell,et al.  Approximation theory and methods , 1984 .

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

[32]  Raytcho D. Lazarov,et al.  Higher-order finite element methods , 2005, Math. Comput..

[33]  D. Wilhelmsen,et al.  A Markov inequality in several dimensions , 1974 .

[34]  Rui Du,et al.  Convergence of the Heterogeneous Multiscale Finite Element Method for Elliptic Problems with Nonsmooth Microstructures , 2010, Multiscale Model. Simul..

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

[36]  I. Babuska,et al.  Acta Numerica 2003: Survey of meshless and generalized finite element methods: A unified approach , 2003 .

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

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