20 research outputs found
A Generalized Multiscale Finite Element Method for the Brinkman Equation
In this paper we consider the numerical upscaling of the Brinkman equation in
the presence of high-contrast permeability fields. We develop and analyze a
robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for
the Brinkman model. In the fine grid, we use mixed finite element method with
the velocity and pressure being continuous piecewise quadratic and piecewise
constant finite element spaces, respectively. Using the GMsFEM framework we
construct suitable coarse-scale spaces for the velocity and pressure that yield
a robust mixed GMsFEM. We develop a novel approach to construct a coarse
approximation for the velocity snapshot space and a robust small offline space
for the velocity space. The stability of the mixed GMsFEM and a priori error
estimates are derived. A variety of two-dimensional numerical examples are
presented to illustrate the effectiveness of the algorithm.Comment: 22 page
Variational multiscale Finite Element Method for flows in highly porous media
We present a two-scale finite element method for solving Brinkman’s and Darcy’s equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes’ equations byWang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy’s equations. In order to reduce the “resonance error” and to ensure convergence to the global fine solution the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems