# Stable Discretizations for Non-Newtonion Flow Models

## Dr. Qingguo Hong

**Dec. 11, 2012, 2:30 p.m. S2 059**

This talk presents a semi-Lagrangian DG method and a semi-Lagrangian nonconforming method that use lower-order polynomial to keep the exact divergence-free of the solution and preserve the positive-definiteness of the conformation tensor regardless of the time and spatial resolutions for the non-Newtonian flow model. Since using lower-order polynomial to approximate velocity, the accuracy of the velocity can match the accuracy of the approximation for the conformation tensor. The discretization schemes are available both in 2-dimensional models and in 3-dimensional models. Because of the divergence-free discrete velocity, we can obtain the robustness of the numerical method by proving the discrete energy estimate. Furthermore, although the existence of the global solution is not clear for the model, but we can prove the existence of the global solution for the discrete problems. Finally, we design a fast solver for the semi-Lagrangian DG discretization, which shows that the semi-Lagrangian DG discretization is solver friendly.