A mixed local discontinuous Galerkin method for a class of nonlinear problems in fluid mechanics

In this paper, we present and analyze a new mixed local discontinuous Galerkin (LDG) method for a class of nonlinear model that appears in quasi-Newtonian Stokes fluids. The approach is based on the introduction of the flux and the tensor gradient of the velocity as further unknowns. In addition, a suitable Lagrange multiplier is needed to ensure that the corresponding discrete variational formulation is well posed. This yields a two-fold saddle point operator equation as the resulting LDG mixed formulation, which is then reduced to a dual mixed formulation. Applying a nonlinear version of the well known Babuška-Brezzi theory, we prove that the discrete formulation is well posed and derive the corresponding a priori error analysis. We also develop a reliable a-posteriori error estimate and propose the associated adaptive algorithm to compute the finite element solutions. Finally, several numerical results illustrate the performance of the method and confirm its capability to localize boundary and inner layers, as well as singularities.