A Multiscale Method for Porous Microstructures

In this paper we develop a multiscale method to solve problems in complicated porous microstructures with Neumann boundary conditions. By using a coarse-grid quasi-interpolation operator to define a fine detail space and local orthogonal decomposition, we construct multiscale corrections to coarse-grid basis functions with microstructure. By truncating the corrector functions we are able to make a computationally efficient scheme. Error results and analysis are presented. A key component of this analysis is the investigation of the Poincareź and inverse inequality constants in perforated domains as they may contain microstructural information. Using first a theoretical method based on extensions of functions and then a constructive method originally developed for weighted Poincareź inequalities, we are able to obtain estimates on Poincareź constants with respect to scale and separation length of the pores. Finally, two numerical examples are presented to verify our estimates.

[1]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[2]  Daniel Peterseim,et al.  Computation of eigenvalues by numerical upscaling , 2012, Numerische Mathematik.

[3]  W. Hackbusch,et al.  Composite finite elements for the approximation of PDEs on domains with complicated micro-structures , 1997 .

[4]  P. Henning,et al.  A localized orthogonal decomposition method for semi-linear elliptic problems , 2012, 1211.3551.

[5]  Patrick Henning,et al.  The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains , 2009, Numerische Mathematik.

[6]  C. Engwer,et al.  An unfitted finite element method using discontinuous Galerkin , 2009 .

[7]  M. Rumpf,et al.  Composite finite elements for 3D image based computing , 2009 .

[8]  S. Sauter,et al.  Extension operators and approximation on domains containing small geometric details , 1999 .

[9]  D. Peterseim.  Variational Multiscale Stabilization and the Exponential Decay of Fine-Scale Correctors , 2015, 1505.07611.

[10]  Andrey L. Piatnitski,et al.  Homogenization: Methods and Applications , 2007 .

[11]  Yalchin Efendiev,et al.  Multiscale Finite Element Methods: Theory and Applications , 2009 .

[12]  Daniel Peterseim,et al.  Oversampling for the Multiscale Finite Element Method , 2012, Multiscale Model. Simul..

[13]  Daniel Peterseim,et al.  Multiscale Petrov-Galerkin method for high-frequency heterogeneous Helmholtz equations , 2015, 1511.09244.

[14]  Daniel Peterseim,et al.  Localization of elliptic multiscale problems , 2011, Math. Comput..

[15]  Daniel Peterseim,et al.  Multiscale Partition of Unity , 2013, 1312.5922.

[16]  Randolph E. Bank,et al.  On the $${H^1}$$H1-stability of the $${L_2}$$L2-projection onto finite element spaces , 2014, Numerische Mathematik.

[17]  C. M. Elliott,et al.  A Finite-element Method for Solving Elliptic Equations with Neumann Data on a Curved Boundary Using Unfitted Meshes , 1984 .

[18]  P. Houston,et al.  hp–Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains , 2014 .

[19]  Giancarlo Sangalli,et al.  Variational Multiscale Analysis: the Fine-scale Green's Function, Projection, Optimization, Localization, and Stabilized Methods , 2007, SIAM J. Numer. Anal..

[20]  Peter Knabner,et al.  Numerik partieller differentialgleichungen : eine anwendungsorientierte Einführung , 2000 .

[21]  Daniel Peterseim,et al.  Two-Level Discretization Techniques for Ground State Computations of Bose-Einstein Condensates , 2013, SIAM J. Numer. Anal..

[22]  Assyr Abdulle,et al.  Localized orthogonal decomposition method for the wave equation with a continuum of scales , 2014, Math. Comput..

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

[24]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[25]  Frédéric Legoll,et al.  An MsFEM Type Approach for Perforated Domains , 2013, Multiscale Model. Simul..

[26]  Axel Målqvist,et al.  Multiscale Methods for Elliptic Problems , 2011, Multiscale Model. Simul..

[27]  Daniel Peterseim,et al.  Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast , 2016, Comput. Methods Appl. Math..

[28]  Robert Scheichl,et al.  Weighted Poincaré inequalities , 2013 .

[29]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .