  The high-order accurate continuous Galerkin finite element method offers
attractive computational efficiency for computational fluid dynamics. A
challenge is however spurious oscillations which result for convection
dominated flows over discontinuities. To derive a continuous Galerkin scheme
for both smooth and discontinuous fields we start by first writing the scheme
in Summation-by-Parts (SBP) form for a single element mesh. Boundary conditions
are applied weakly via Simultaneous-Approximation-Terms (SAT) and Gauss-Labotto
quadrature employed in the interest of computational efficiency. We then show
that the stable single element baseline scheme in SBP-SAT form extends
trivially to a provably stable multi-element formulation. Next, we develop
provably stable element based Galerkin-weighted artificial dissipation
operators to deal with spurious oscillations over shocks while retaining high
order accuracy elsewhere. The resulting scheme achieves super-convergence with
accuracy of Order(p+2) when using p^th order Lagrange polynomials for smooth
fields. The developed dissipation operators furnish WENO like behaviour over
discontinuities while retaining high order accuracy elsewhere for both linear
and non-linear wave propagation problems.
