Solution of Steady and Transient Advection Problems using an h-Adaptive Finite Element Method

A standard h-adaptive finite element procedure based on a-posteriori error-estimation is described. The pure advection equation is solved (in both steady and transient states) using the SUPG (streamline upwind Petrov-Galerkin) formulation of the finite element method. Applied to standard benchmark problems (of uniform flow advecting discontinuous functions) the SUPG method on its own is insufficient to resolve the sharp discontinuities present when used with a uniform mesh of insufficient refinement. The amount of false diffusion is also seen to be related to the degree of mesh refinement. An iterative h-adaptive procedure used in combination with the SUPG formulation (with a discontinuity capturing term) produces a near perfect solution of the steady state benchmark problem. The transient benchmark problem (rotating cosine hill) shows that the adaptive Galerkin FEM in combination with central difference time integration produces solutions indistinguishable from the corresponding adaptive SUPG solution. This result clearly indicates that at least for transient problems correct mesh refinement with GFEM can overcome the wiggle problems associated with using GFEM with central difference time integration for advection problems. So in effect adaptivity in addition to providing the expected benefits of selective mesh refinement also acts as a wiggle suppressent for an otherwise highly oscillatory GFEM/Central Difference combination. This result is even more interesting in view of the fact that when SUPG is used it does not improve significantly on the quality of the adaptive GFEM/CD solution.