A nonlinear subgrid method for advection–diffusion problems

This work presents a general framework for approximating advection–diffusion equations based on principles of scale separation. A two-level decomposition of the discrete approximation space is performed and the local problem is modified to capture both local and nonlocal discontinuities. The new feature is the local control resulting from decomposing the velocity field into the resolved and unresolved scales and requiring the satisfaction of the discrete model problem at the element level for a minimum kinetic energy associated to the unresolved scales. This procedure leads to a nonlinear subgrid model that acts only on the unresolved scales but does not require any tuned-up parameter. It can be considered a self-adaptive method such that the amount of the subgrid viscosity is automatically introduced according to the residual of the resolved scale at element level.