Computation facilitates to understand phenomena and processes from science and engineering; we no longer need to depend only on theory and experiment. Computational Science and Engineering (CSE) is a rapidly developing multidisciplinary area using computational mathematics in the fields of science and engineering. CSE focuses on modeling-computer simulation-visualization, based on applied mathematics. We aim to provide problem-solving methodologies and robust tools for numerical simulation.
In this talk, we present our recent efforts for developing a robust numerical scheme for various problems including the Darcy and the Navier-Stokes equations. Hybrid discontinuous Galerin methods (HDG) was first designed and proposed by Y. Jeon and myself to study the Darcy equation. We further develop the method and provide arbitrary-order, locally conservative, stabilized formulation for Navier-Stokes problems. Several numerical results are presented to test the performance of the algorithm and to validate the theory developed. For stationary incompressible Navier-Stokes equations, numerical results for the lid-driven cavity problem are presented with Reynolds numbers up to 21000, and compared with existing results. This is a joint work with Y. Jeon and D. Shin.