A three-scale offline–online numerical method for fluid flow in porous media

A new multiscale method combined with model order reduction is proposed for flow problems in three-scale porous media. We derive an effective three-scale model that couples a macroscopic Darcy equation, a mesoscopic Stokes-Brinkman equation, and a microscopic Stokes equation. A corresponding three-scale numerical method is then derived using the finite element discretization with numerical quadrature, where the macroscopic and mesoscopic permeability is upscaled at quadrature points from mesoscopic and microscopic problems, respectively. The computational cost of solving numerous mesoscopic and microscopic flow problems is further reduced by applying a Petrov–Galerkin reduced basis method at the mesocopic and microscopic scales. As there is no natural way to obtain an affine decomposition of the mesoscopic problems, which is instrumental for the efficiency of the model order reduction, we derive a mesoscopic solver that makes use of empirical interpolation techniques. A priori and a posteriori error estimates are derived for the new method that is also tested numerically to corroborate the theoretical convergence rates and illustrate its efficiency.

MATHICSE Technical Report : Multiscale methods and model order reduction for flow problems in three-scale porous media

Assyr Abdulle, Ondrej Budác

A new multiscale method combined with model order reduction is proposed for flow problems in three-scale porous media. We derive an effective three-scale model that couples a macroscopic Darcy equation, a mesoscopic Stokes-Brinkman equation, and a microscopic Stokes equation. A corresponding three-scale numerical method is then derived using the finite element discretization with numerical quadrature, where the macroscopic and mesoscopic permeability is upscaled at quadrature points from mesoscopic and microscopic problems, respectively. The computational cost of solving numerous mesoscopic and microscopic flow problems is further reduced by applying a Petrov–Galerkin reduced basis method at the mesocopic and microscopic scales. As there is nonatural way to obtain an affine decomposition of the mesoscopic problems, which is instrumental for the efficiency of the model order reduction, we derive a mesoscopic solver that makes use of empirical interpolation techniques. A priori and a posteriori error estimates are derived for the new method that is also tested numerically to corroborate the theoretical convergence rates and illustrate its efficiency.

MATHICSE2016

MATHICSE Technical Report: A discontinuous Galerkin reduced basis numerical homogenization method for fluid flow in porous media

Assyr Abdulle, Ondrej Budác

We present a new conservative multiscale method for Stokes flow in heterogeneous porous media. The method couples a discontinuous Galerkin finite element method(DG-FEM) at the macroscopic scale for the solution of an effective Darcy equation with a Stokes solver at the pore scale to recover effective permeabilities at macroscopic quadrature points. To avoid costly computation of numerous Stokes problems through-out the macroscopic computational domain, the pore geometry is parametrized and a model order reduction algorithm is used to select representative microscopic geometries. Accurate Stokes solutions and related permeabilities are obtained for these representative geometries in an offline stage. In an online stage, the DG-FEM is computed with permeabilities recovered at the desired macroscopic quadrature points from the precomputed Stokes solutions. The multiscale method is shown to be mass conservative at the macro scale and the computational cost for the online stage is similar to the cost of solving a single scale Darcy problem. Numerical experiments for two and three dimensional problems illustrate the efficiency and the performance of the proposed method.

MATHICSE2015

Fully discrete analysis of the heterogeneous multiscale method for elliptic problems with multiple scales

Assyr Abdulle, Yun Bai

A fully discrete analysis of the finite element heterogeneous multiscale method (FE-HMM) for elliptic problems with N+1 well-separated scales is discussed. The FE-HMM is a numerical homogenization method that relies on a macroscopic scheme (macro FEM) for the approximation of the effective solution corresponding to the multiscale problem. The effective data are recovered from micro scale computation (micro FEM) on sampling domains located at appropriate quadrature points of the macroscopic mesh. At the macroscopic level, the numerical method can be seen as an FEM with numerical quadrature for a modified effective problem, hence variational crimes are made when designing this method. Up to now, the method has been analysed for two scales and the micro FEM was assumed to be conforming. For more than two scales, variational crimes are committed also at the intermediate (meso) scales and the effective data of the macroscopic scheme are obtained from a cascade of FEMs with numerical integration, which require a careful analysis. Numerical experiments for three-scale problems illustrate the theoretical convergence rates.

Oxford Univ Press2015