We are interested in the numerical simulation of continuum mechanics systems by means of finite element schemes for appropriate relaxation models, which formally arise as hyperbolic singular perturbations of the original equations. Analytical investigations indicate that relaxation provides a dissipative mechanism against the destabilizing effect of nonlinear terms, as well as damping effects on oscillations when assisted by nonlinear response. From a numerical point of view, the utilization of relaxation methods result in discrete systems with linear principal part and the nonlinearity confined to zero order source terms, which is an advantage especially for simulating conservation laws with nonlinear fluxes because it avoids the need for complex Riemann solvers. In addition, the relaxation regularization may prevent from using extra stabilization techniques such as limiters or shock capturing operators. Finite element schemes use piecewise polynomials of arbitrary degree and exhibit higher order consistency error on unstructured meshes, thus producing efficient numerical solvers for multi-dimensional problems. The natural framework for relaxation approximations is offered by the theory of materials with internal variables, which describes the diffusive stress relaxation of the system of visco-elastodynamics through a small dissipative correction as part of a wave operator with finite speed of propagation. In terms of industrial applications, the behavior of viscoelastic materials has a profound influence on their performance to achieve particular goals: for example, viscoelastic shoe insoles are useful in reducing mechanical shocks transmitted to the bones and joints, and the recent introduction of memory foam for mattresses and pillows is another important exploitation. The objective of the study currently conducted is to check finite element relaxation methods for the equations of elastodynamics combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. Stability properties of the schemes are derived from estimating suitable modifications of the standard energy functional, that is suggested by the presence of relaxation terms, and these techniques are also applied to prove error estimates for deducing the convergence to the (smooth) solutions of the equations. A similar approach has also been developed for the numerical modelling of the incompressible Navier-Stokes equations, in two space dimensions.