  Simulation of multiscale problems remains a challenge due to the disparate
range of spatial and temporal scales and the complex interaction between the
resolved and unresolved scales. This work develops a coarse-grained modeling
approach for the Continuous Galerkin discretizations by combining the
Variational Multiscale decomposition and the Mori-Zwanzig (M-Z) formalism. An
appeal of the M-Z formalism is that - akin to Greens functions for linear
problems - the impact of unresolved dynamics on resolved scales can be formally
represented as a convolution (or memory) integral in a non-linear setting. To
ensure tractable and efficient models, Markovian closures are developed for the
M-Z memory integral. The resulting sub-scale model has some similarities to
adjoint stabilization and orthogonal subscale models. The model is made
parameter free by adaptively determining the memory length during the
simulation. To illustrate the generalizablity of this model, it is employed in
coarse-grained simulations for the one-dimensional Burgers equation and in
incompressible turbulence problems.
