Enforcement of constraints and maximum principles in the variational multiscale method

We present a new theoretical framework for the enforcement of constraints in variational multiscale (VMS) analysis. The theory is first presented in an abstract operator format and subsequently specialized for the steady advection–diffusion equation. The approach borrows heavily from results in constrained and convex optimization. An exact expression for the fine-scales is derived in terms of variational derivatives of the constraints, Lagrange multipliers, and a fine-scale Green’s function. The methodology described enables the development of numerical methods which satisfy predefined attributes. A practical and effective procedure for solving the steady advection–diffusion equation is presented based on a VMS-inspired stabilized method, weakly enforced Dirichlet boundary conditions, and enforcement of a maximum principle and conservation constraint.