题目：Recent developments on direct discontinuous Galerkin methods s
报告人：阎珏教授 (Iowa State University)
摘要：We first introduce the direct discontinuous Galerkin (DDG) method and its variations, namely the DDGIC and symmetric DDG methods. Compared to the leading diffusion DG method solvers like the interior penalty method (SIPG), we find out our diffusion solver the DDG methods have many advantages. While the SIPG method needs polynomial degree dependent large enough penalty coefficient to stabilize the scheme, numerically we observe small fixed constant penalty coefficient is enough for the DDGIC method to obtain optimal convergence. Under the topic of maximum principle, DDG methods numerical solution can be proved to satisfy strict maximum principle even on unstructured triangular meshes with at least third order of accuracy.
Recently we develop DDG methods to solve Keller-Segel Chemotaxis equations. Different to available DG methods or other numerical methods in literature, we introduce no extra variable to approximate the chemical density gradients and we solve the system directly. With $P^k$ polynomial approximations, we observe no order loss for the density variable. The reason behind is that the DDGIC or symmetric DDG methods have the hidden super convergence property on its approximation to the solution gradients. With Fourier (Von Neumann) analysis technique, we prove the DDG solution’s spatial derivative is super convergent with at least (k+1)th order under moment format or in the weak sense. Notice that we do not have super convergence for the SIPG method. We show the cell density approximations are strictly positive with at least third order of accuracy.