A local discontinuous Galerkin (LDG) finite element method for the solution of a hyperbolic--elliptic system modeling the propagation of phase transition in solids and fluids is presented. Viscosity and capillarity terms are added to select the physically relevant solution. The $L^2$--stability of the LDG method is proven for basis functions of arbitrary polynomial order. In addition, using a priori error analysis, we provide an error estimate for the LDG discretization of the phase transition model when the stress--strain relation is linear, assuming that the solution is sufficiently smooth and the system is hyperbolic. Also, results of a linear stability analysis to determine the time step are presented. To obtain a reference exact solution we solved a Riemann problem for a trilinear strain--stress relation using a kinetic relation to select the unique admissible solution. This exact solution contains both shocks and phase transitions. The LDG method is demonstrated by computing several model problems representing phase transition in solids and in fluids with a Van der Waals equation of state. The results show the convergence properties of the LDG method.