SUPG-based stabilized finite element computations of convection-dominated 3D elliptic PDEs using shock-capturing. (English) Zbl 07890871
Summary: This study is interested in stabilized finite element computations of advection-dominated convection-diffusion-type partial differential equations. The governing equations are assumed to be defined on the 3D unit cube and stationary. Towards that end, we stabilize the standard (Bubnov-) Galerkin finite element method (GFEM), which typically suffers from yielding numerical approximations polluted with node-to-node nonphysical oscillations in convection dominance, by employing the streamline-upwind/Petrov-Galerkin (SUPG) formulation. In order to achieve better shock representations near sharp layers, the SUPG-stabilized formulation is also enhanced with a simple and residual-based shock-capturing mechanism, the so-called YZ\(\beta\) technique. Several test computations are performed to assess the efficiency and robustness of the proposed formulation. The test problems are examined under more challenging conditions than those previously studied in the literature, i.e., for flows substantially dominated by convection phenomena. The numerical results and comparisons demonstrate that the SUPG-YZ\(\beta\) combination significantly eliminates both globally spread and localized numerical instabilities. Beyond that, the proposed formulation accomplishes that by using only linear elements on relatively coarser meshes and without making use of any adaptive mesh strategies.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76R99 | Diffusion and convection |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
convection-diffusion equation; streamline-upwind/Petrov-Galerkin formulation; GMRES search technique; ILU preconditioning; Newton-Raphson iterationReferences:
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