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 ste ady and transient slates) using the SUPG (streamline upwind Petrov-Galerkin ) formulation of the finite element method, Applied to standard benchmark p roblems (of uniform flow advecting discontinuous functions) the SUPG method on its own is insufficient to resolve the sharp discontinuities present wh en 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 formulatio n (with a discontinuity capturing term) produces a near perfect solution of the steady state benchmark problem. The transient benchmark problem (rotat ing cosine hill) shows that the adaptive Galerkin FEM in combination with c entral difference time integration produces solutions indistinguishable fro m the corresponding adaptive SUPG solution. This result clearly indicates t hat at least for transient problems correct mesh refinement with GFEM can o vercome the wiggle problems associated with using GFEM with central differe nce lime integration for advection problems. So in effect adaptivity in add ition to providing the expected benefits of selective mesh refinement also acts as a wiggle suppressent for an otherwise highly oscillatory GFEM/Centr al 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 qu ality of the adaptive GFEM/CD solution.