The discontinuous Galerkin finite element method for two-dimensional convection-diffusion problems on triangular meshes
Problems involving convection and diffusion arise in several important applications throughout science and engineering, including meteorology, oceanography, gas dynamics, acoustics, biological and chemical transport, elasticity and plasticity, magneto-hydrodynamics, electricity and magnetism, and microelectronics. Designing and developing efficient, accurate, and robust numerical methods to address these problems has survived as a classical challenge since the earliest days of digital computation. The basic reasons for the difficulty involve (i) the development and tracking of discontinuities (for convection problems) or sharp transition layers (for convection-diffusion problems) and (ii) the generation of numerical solutions that fail to satisfy physical (e.g., entropy) principles.
Classical numerical methods such as finite difference, finite volume, and finite element schemes have been developed to overcome these difficulties to a certain extent. However, they each suffer limitations that can potentially be overcome by the discontinuous Galerkin (DG) method, which combines the best features of each of the more traditional approaches. DG method is becoming important technique for the computational solution of many differential equations. The DG method is a class of finite element methods which use completely discontinuous functions as the numerical solutions and test functions. DG method has been under rapid development in recent years as one of the powerful numerical method to solve various partial differential equations (PDEs). Stable and convergent DG methods have been designed in recent years for various types of PDEs, including hyperbolic, parabolic and elliptic equations. DG methods have found rapid applications recently, in such diverse areas as aeroacoustics, electromagnetism, gas dynamics, granular flows, meteorology, oceanography, semiconductor device simulation, turbulent flow, weather forecasting, among many others.
The local discontinuous Galerkin (LDG) finite element method is an extension of the discontinuous Galerkin (DG) method aimed at solving ordinary and partial differential equations containing higher than first-order spatial derivatives. In this project, we analyze the superconvergence properties of the LDG method applied to two-dimensional convection-diffusion problems on triangular meshes. Finally, we will use the superconvergence results to construct simple, efficient, and asymptotically correct a posteriori error estimates by solving local problems with no boundary conditions on each element. A posteriori error estimates are needed to guide adaptive enrichment and to provide a measure of solution accuracy for any numerical method. Numerical experiments will be presented to validate the theoretical results.
MATH 3300 and MATH 4330 are required. You will be using Matlab.