We construct parallel finite element methods for the solution of hyperbolic
conservation laws in one and two dimensions. Spatial discretization is
performed by a discontinuous Galerkin finite element method using a basis of
piecewise Legendre polynomials. Temporal discretization utilizes a
Runge-Kutta method. Dissipative fluxes and projection limiting prevent
oscillations near solution discontinuities. A posteriori estimates of spatial
errors are obtained by a *p*-refinement technique using superconvergence
at Radau points. The resulting method is of high order and may be
parallelized efficiently on MIMD computers. We compare results using
different limiting schemes and demonstrate parallel efficiency through
computations on an nCUBE/2 hypercube. We also present results using adaptive
*h-* and *p*-refinement to reduce the computational cost of the
method.