Abstract: In this talk, we present error analysis and numerical results showing optimal convergence and superconvergence properties of the hybridizable discontinuous Galerkin (HDG) methods. First, we study the convection-diffusion equations with variable-degree approximations on nonconforming meshes. Our results hold for any (bounded) irregularity index of the nonconformity of the mesh, and can be extended to hypercubes. Second, we design and analyze the first HDG methods for stationary, third-order linear equations in one-space dimension. 13 methods are analyzed in a unified setting. They all provide superconvergent approximations to the exact solution u and its two derivatives. Numerical results validate our theoretical findings.

Biography: Yanlai Chen is an Assistant Professor of Mathematics at UMass Dartmouth. Before joining UMassD, he was a postdoctoral research associate at Brown University. He received his Ph.D. in Mathematics and M.S. in computer science from University of Minnesota in 2007, and B.S. degree in Mathematics from University of Science and Technology of China (USTC) in 2002. Dr. Chen’s research area is numerical analysis and scientific computing, in particular discontinuous Galerkin finite element methods, hybridizable discontinuous Galerkin methods, reduced basis method and reduced basis element method. His research is currently supported by NSF.

The development of a mathematical procedure by the MSEAS group to coordinate the movements of AUVs in the extremely dynamic coastal environment in which the Pioneer Array will be situated is detailed in a news item on the new MIT oceans.mit.edu web site. The news article describes the coordination between the MSEAS group and the LAMSS group headed by Prof. Henrik Schmidt in the building of the cyberinfrastructure necessary to support the communications between sensors and models.

Two papers by Sondergaard and Lermusiaux have been accepted for publication in Monthly Weather Review. These papers are on Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations. The first describes the theory and scheme, while the second discusses applications.