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We propose a class of finite element schemes for systems of hyperbolic conservation laws that are based on finite element discretizations of appropriate relaxation models. We consider both semidiscrete and fully discrete finite element schemes and show that the schemes are stable and, when the compensated compactness theory is applicable, do converge to a weak solution of the hyperbolic system. The schemes use piecewise polynomials of arbitrary degree and their consistency error is of high order. We also prove that the rate of convergence of the relaxation system to a smooth solution of the conservation laws is of order $O(\eps )$.