Summary: HOMOGENIZATION-BASED MIXED MULTISCALE FINITE
ELEMENTS FOR PROBLEMS WITH ANISOTROPY
Abstract. Multiscale finite element numerical methods are used to solve flow problems when
the coefficient in the elliptic operator is heterogeneous. A popular mixed multiscale finite element
has basis functions which can be defined only over pairs of elements, so we call it a "dual-support"
element. We show by example that it can fail to reproduce constant flow fields, and so fails to converge
in any meaningful way. The problem arises when the coefficient is an anisotropic tensor. A new
approach to multiscale finite elements based on the microscale structure theory of homogenization is
presented to avoid the problems with anisotropy. Five numerical test cases are presented to evaluate
and contrast the methods. The first involves anisotropy, and the second is similar in that, although
it has an isotropic coefficient, its heterogeneity leads to an anisotropic homogenized coefficient.
As expected, the popular method has difficulty--while the new method shows no difficulty--with
either anisotropy or macroscale implied anisotropy. The final three tests involve heterogeneous and
channelized cases, and features of the new method are shown to be important for good approximation.
Finally, for a two-scale coefficient, a proof of convergence is presented for standard mixed multiscale
finite elements that reduces to four simple steps. From its simplicity, one can easily see that the
popular elements fail only the step related to the counterexample, and we conjecture, but do not
prove, that the new homogenization-based elements converge.
Key words. elliptic, heterogeneous, mixed method, convergence, dual-support elements, dual-