## A Unified Framework for Multiscale Modeling using the Mori-Zwanzig Formalism and the Variational Multiscale Method. (arXiv:1712.09669v1 [math.NA])

We describe a paradigm for multiscale modeling that combines the Mori-Zwanzig
(MZ) formalism of Statistical Mechanics with the Variational Multiscale (VMS)
method. The MZ-VMS approach leverages both VMS scale-separation projectors as
well as phase-space projectors to provide a systematic modeling approach that
is applicable to non-linear partial differential equations. Spectral as well as
continuous and discontinuous finite element methods are considered. The
framework leads to a formally closed equation in which the effect of the
unresolved scales on the resolved scales is non-local in time and appears as a
convolution or memory integral. The resulting non-Markovian system is used as a
starting point for model development. We discover that unresolved scales lead
to memory effects that are driven by an orthogonal projection of the
coarse-scale residual and inter-element jumps. It is further shown that an
MZ-based finite memory model is a variant of the well-known
adjoint-stabilization查看全文