On the stability of low-order continuous and discontinuous mixed finite element methods in viscoeleastic fluid mechanics

In recent years a lot of research has been performed on the development of numerical tools for viscoelastic flow simulations. The nature of the governing equations, i.e. constitutive equations and conservation of mass and momentum, requires that special attention has to be paid to the numerical solution algorithm. For instance, a major problem that needs to be resolved is the loss of stability of standard Galerkin solution procedures with increasing Weissenberg numbers. To overcome this problem, several stabilising techniques have been proposed such as application of Petrov-Galerkin weighting or discontinuous methods, as is common for finite difference schemes. Another challenging task is created by the complex rheology of viscoelastic fluids. Since, realistic analysis of polymer flows compels the use of multiple viscoelastic modes, a very large number of global degrees of freedom is obtained when mixed methods are applied. This, in turn, requires a highly efficient solution strategy. The mixed finite element method that will be presented is based on a new Discontinuous Galerkin technique to obtain local discontinuous approximations of all independent variables. Although closely related to the mortar element method, this discontinuous technique allows for jumps across the element boundaries and does not require auxiliary variables to enforce continuity. Due to the local support of the polynomial basis, the method is suited for an element-by-element time marching algorithm yielding an efficient handling of the multitude of unknowns. Furthermore, the local character of the bases implies that the method is highly flexible and suited for adaptive polynomial refinement and non-matching grids.