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