The USFEM Method And Duan's Method For The Convection-Reaction-Diffusion Equations On Isotropic And Anisotropic Meshes

In this paper, firstly, we give a new proof of the numerical stability and conver-gence of the USFEM method(the Unusual Stabilized Finite Element Method)derived byL.P.Franca and F.Valentin[5] on isotropic meshes.Secondly, we extend both methods toanisotropic meshes following the work by Stefano Micheletti, Simona Perrotto, MarcoPicasso in the paper [12],and with the relationship between USFEM and GalerkinMethods Enriched with Bubble Functions together. Finally, we study properties ofthe element and face bubble on anisotropic meshes, and then we extend G.Kunert'stricks when using general reference triangle or tetrahedron. Using G.Kunert's tricks,westudy the stability and efficiency of Z-Z like anisotropic posteriori error estimator onseveral models. Especially, we study the stability and efficiency of Z-Z like anisotropicposteriori error estimator when using G.Kunert's reference triangle.