Stabilized FEM's and residual-free bubbles for convection diffusion problems: anisotropic meshes and adaptivity

Dr. Andrea Cangiani

Sept. 1, 2005, 11 a.m. HF 9904

The theory of stabilized methods (e.g. SUPG, GLS) for the numerical solution of linear convection-diffusion problems is well established. Recently, attempts have been made to derive stabilized methods from fundamental principles to better understand their stabilization mechanisms and possible generalizations. This goal led to the development of the residual-free bubble (RFB) method. This is based on the idea of enriching the finite element space and is justified by the fact that the solution of convection-dominated-diffusion problems present multi-scale behaviour (e.g. thin layers). We shall present our work on the analysis of the RFB method concentrating on anisotropic meshes and mesh adaptivity. As a byproduct of such analyses, we derived a theoretically optimal value for the SUPG stabilization parameter on anisotropic meshes and obtained a new local indicator that detects in which portions of the mesh stabilization is really necessary. Finally, we shall present a new algorithm (RFBe) based on combining RFB with exponential fitting techniques.