A generalized multiscale finite element method for the Brinkman equation

Juan Galvis ^[1] ; Guanglian Li ^[2] ; Ke Shi ^[2]
1. [1] Universidad Nacional de Colombia
  
  Universidad Nacional de Colombia
  
  Colombia
2. [2] Texas A&M University
  
  Texas A&M University
  
  Estados Unidos
Localización: Journal of computational and applied mathematics, ISSN 0377-0427, Vol. 280, Nº 1 (15 May 2015), 2015, págs. 294-309
Idioma: inglés
DOI: 10.1016/j.cam.2014.11.038
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Resumen
- 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 two dimensions. 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.