Abstract. 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 solution. The computational performance of these methods is demonstrated by considering scalar problems and the system of elastodynamics.Mathematics Subject Classification. 35L65, 65M60, 65M50, 82C40.

In this work we consider a new class of Relaxation Finite Element schemes for Conservation Laws, with more stable behavior on the limit area of the relaxation parameter. Combine this scheme with an efficient adapted spatial redistribution process, considered also in this work, we form a robust scheme of controllable resolution. The results on a number of test problems show that this scheme can produce entropic-approximations of high resolution even on the limit of the relaxation parameters. Since on the limit the scheme lack of the relaxation mechanism, we experimentally conclude that the proposed spatial redistribution can be a stabilization mechanism by its own for computational solutions of CL problems.

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(ε).