A few comments on your "high-speed" compression test. 

A velocity of 1 m/s is not very high speed relative to material properties.  It is high-speed relative to the typical testing speeds of tensile testing machines such as Instron.

You are using a Plasticity material property (multilinear isotropic hardening), which is Rate Independent. In other words, this material will behave the same at any strain rate. Therefore you would expect to get the same stress-strain curve at different velocities, however you haven't changed the velocity.

Rate Dependent material models, either viscoelastic or viscoplastic, change their response with strain rate (velocity) or in some other way depend on time.

The boundary condition of -1 m/s velocity at t=0 to the top surface creates an infinite acceleration. The shock of that causes oscillations to begin and over time they dissipate due to damping. It seems the infinite acceleration shock causes a larger shock wave to bounce around inside a mesh of smaller elements than larger elements. 

It would be better to ramp the velocity up from zero at t=0 to -1 m/s at some later time. This will create a finite acceleration at t=0 which will greatly reduce the oscillations at all element sizes.