Over the past decade, several closure and stabilization strategies have been developed to tackle the ROM inaccuracy in the convection-dominated, under-resolved regime, i.e., when the number of degrees of freedom is too small to capture the complex underlying dynamics.
Since the realization in 1907 by Hermann Minkowski that spacetime is a four-dimensional structure and his key invention of the light cone and the subsequent development by his former student Albert Einstein and Marcel Grossmann of the theory of gravity, the properties of wave propagation in spacetime have been under intense scrutiny.
In this work we propose and analyze an augmented mixed formulation for the time-dependent Brinkman-Forchheimer equations written in terms of vorticity, velocity and pressure.
The weak formulation is based on the introduction of suitable least squares terms arising from the incompressibility condition and the constitutive equation relating the vorticity and velocity. We establish existence and uniqueness of a solution to
One of the biggest open problems in mathematical physics has been the problem of formulating a complete and consistent theory of quantum gravity. Some of the core technical and epistemological difficulties come from the fact that General Relativity is fundamentally a geometric theory and, as such, it ought to be invariant under change of coordinates by the arbitrary element of the diffeomorphism group Diff(M) of the ambient manifold M.
We discuss several recent results for Navier-Stokes equation in EMAC form, including longer time accuracy through an improved Gronwall estimate, a lower bound on error for forms that do not conserve momentum and angular momentum, exact local discrete balances of momentum and angular momentum with EMAC, and lastly the importance of consistency in ROMs for NSE using EMAC.