An SBP-SAT Continuous Galerkin Finite Element Formulation for Smooth and Discontinuous Fields

Abstract

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.