Multi-scale discontinuous Galerkin method for solving elliptic problems with curvilinear unidirectional rough coefficients (Q474972)

The authors propose a multi-scale discontinuous Galerkin method for the second-order boundary value problem \[ -\nabla \cdot (A(x) \nabla u) = f(x) \;\mathrm {in}\;\Omega, \] with \(u=0\) on \(\partial \Omega\), and \(A(x)\) contains highly oscillatory functions. They construct a special approximation space to cope with the curvilinear unidirectional rough coefficients and prove optimal error estimates for a second-order approximation space regardless of the fine scales.