An Optimization Based Coupling Method For Multiscale Problems

A new multiscale coupling method is proposed for elliptic problems with highly oscillatory coefficients with a continuum of scales in a subset of the computational domain and scale separation in complementary regions of the computational domain. A discontinuous Galerkin (DG) finite element heterogeneous multiscale method (FE-HMM) is used in the region with scale separation, while a continuous standard finite element method is used in the region without scale separation. The use of a DG-FE-HMM method allows for a flexible meshing of the different models in the overlapping region. The unknown boundary conditions at the interfaces are obtained by minimizing the error of the two models in the overlapping region. We prove the well-posedness of both the continuous and discrete coupling problems and establish convergence of the multiscale method towards the fine scale solution. Since in the region with scale separation we obtain an approximation at a cost independent of the smallest scale in the problem, the computational cost of the multiscale method is significantly smaller than a fine scale solver over the whole computational domain, while the algorithm allows us to treat situations for which standard numerical homogenization methods do not apply.