On some recent adventures into improved finite element CFD methods for convective transport

The quest continues for computational fluid dynamics (CFD) algorithms that are accurate and efficient for convection-dominated applications including shocks, travelling fronts and wall-layers. The boundary-value 'optimal' Galerkin weak statement invariably requires manipulation, either in the test space or in an augmented form for the conservation law system, to handle the disruptive character introduced by the discretized first-order convection term. An incredible variety of methodologies have been derived and examined to address this issue, in particular seeking achievement of accurate and monotone discrete approximate solutions in an efficient implementation. The UTK CFD research group continues its search on examining the breadth of approaches leading to development of a consistent, encompassing theoretical statement exhibiting quality performance. Included herein are adventures into generalized Taylor series (Lax-Wendroff) methods, characteristic Euler flux resolutions, sub-grid embedded high-degree Lagrange bases with static condensation, and assembled-stencil Fourier analysis optimization for finite element weak statement implementations. For appropriate model problems, including steady convection-diffusion and pure unsteady convection, and benchmark Navier-Stokes definitions, recent advances have lead to candidate accurate monotone methods with linear basis efficiency. This contribution highlights the theoretical developments, and presents quantitative documentation of achievable high quality solutions.