We propose and study semidiscrete and fully discrete
finite element schemes based on appropriate relaxation models for
systems of Hyperbolic Conservation Laws.
These schemes are using piecewise polynomials of arbitrary degree and
their consistency error is of high order.
The methods are combined with an adaptive strategy that yields
fine mesh in shock regions and coarser mesh in the smooth parts of the
The computational performance of these methods is demonstrated by considering
scalar problems and the system of elastodynamics.