Stabilisation using bubbles in FE time dependent problem with application in forming

Abstract : A simple and unified approach is described to stabilize FE on unstructured meshes for a large panel of problems. A variational multiscale approach may be used as a guideline to obtain stability. The simple P1 continuous approximation is systematically augmented with a bubble function inside elements. The bubble is condensed giving rise to supplementary terms in the varitional formulation. Bubble addition comes from the mixed formulation of incompressible flow problem. It is also introduced to retrieve the SUPG method for convective equation, giving rise also in a more general framework to the RFB methods. Different applications are proposed: the incompressible Navier Stokes equation, the interface capturing by Level set method and also a space time continuous P1 Finite Element method working with a full unstructured mesh both in space and time.