Homogenization-Based Mixed Multiscale Finite Elements for Problems with Anisotropy

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.