Verification of a three-dimensional unstructured finite element method using analytic and manufactured solutions

Abstract

We report on the verification of a three-dimensional unstructured finite element method applicable to compressible fluid dynamics and diffusion problems. Our verification methodology uses a combination of analytic and manufactured solutions to formally measure convergence rates in global error for both shock-dominated flows and smooth problems. In addition we measure the global error in vorticity, which should converge at reduced-order relative to the velocity solution. The numerical method under investigation is an edge-based Finite Element formulation on linear tetrahedra with a parabolic MUSCL reconstruction for the advective fluxes. The scheme is nominally second-order accurate on smooth flows. For diffusion problems the formulation also is nominally second-order accurate.Using global error analysis we measure convergence rates of 0.8–1.0 for shock-dominated problems and 1.5–2.4 for smooth problems. Calculations with Adaptive Mesh Refinement (AMR) are observed to produce errors comparable to finer mesh simulations but at significantly reduced computational cost. A convergence rate of 2.2 also is observed for a simplified diffusion problem. Examples of how these studies can inform simulation practices are provided.