The Development of the Multiscale Hybrid-Hybrid-Mixed method

Abstract

This work develops the Multiscale Hybrid-Hybrid Mixed method - MH2 M. This is a finite element method that efficiently solves elliptic partial differential equations with multiscale heterogeneous coefficients. The starting point is the Three-field domain decomposition formulation, which searches a function, defined within each subdomain, and two Lagrange multipliers: the flow and trace of the function posed on interfaces. This setting allows different discretizations in each subdomain, as well as the use of different numerical methods to solve local problems. After the decomposition of functional spaces and two static condensations, the MH2 M method arises by solving independent local Neumann problems in parallel. It results that the method solves an elliptic global problem posed at interfaces instead of the more complicated three-field formulation. In addition to the lower computational cost, the use of iterative methods such as the conjugate gradient is possible. A proper compatibility condition enables a discretization using non-matching grids, preserving stability. Finally, we establish error estimates for a pair of compatible finite element spaces.