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.

翻译：高阶精确连续伽辽金有限元方法为计算流体动力学提供了颇具吸引力的计算效率。然而，对于对流主导的间断流动，会产生虚假振荡这一挑战。为推导适用于光滑场与非连续场的连续伽辽金格式，我们首先针对单个单元网格，将格式写成求和-分部（SBP）形式。通过同时逼近项（SAT）弱施加边界条件，并采用高斯-洛巴托求积以提高计算效率。随后证明，稳定的单单元基线SBP-SAT格式可简单扩展为可证明稳定的多单元公式。接着，我们开发了基于单元的可证明稳定的伽辽金加权人工耗散算子，用于处理激波处的虚假振荡，同时在其他区域保持高阶精度。当使用p阶拉格朗日多项式处理光滑场时，所得格式可实现阶(p+2)的超收敛精度。所开发的耗散算子在线性与非线性波传播问题中，能在间断处呈现类似WENO的行为，同时在其他区域保持高阶精度。