A multiscale finite element method for an elliptic distributed optimal control problem with rough coefficients and control constraints (Q6592270)

A multiscale finite element method is constructed and analyzed for an elliptic distributed optimal control problem with pointwise control constraints, where the state equation has rough coefficients. The DD-LOD multiscale finite element method is one of the simplest multiscale finite element methods in terms of construction and analysis. It is shown that the approximate solution obtained by the DD-LOD multiscale finite element method on the coarse mesh is, up to an \(\mathcal{O} (H^2 +\rho)\) term for the \(L_2\) error and an \(\mathcal{O} (H +\rho)\) term for the energy error, as good as the approximate solution obtained by a standard finite element method on a fine mesh. \(\mathcal{T}_\rho\) be a simplicial/quadrilateral triangulation of \(\Omega \) with mesh size \(\rho\) is a simplicial/quadrilateral triangulation of \(\Omega\), and simplicial/quadrilateral triangulation \(\mathcal{T}_H\) of \(\Omega\) is a refinement \(\mathcal{T}_H\) (\(h \ll H\)) of \(\mathcal{T}_H\). The performance of the DD-LOD method is as good as standard finite element methods for smooth problems. Numerical results are presented for two examples, one with highly heterogeneous coefficients and one with highly oscillatory coefficients.