Grant: Discovery Grants Program - Individual

Numerical analysis is the area of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically the problems of continuous mathematics. Such problems occur throughout the natural and social sciences, psychology, engineering, medicine, and business. In the 21st century, the life sciences and even the arts have adopted elements of scientific computations. We use a wide variety of methods, but finite element methods are some of the most powerful methods in modern computational mathematics.One of the prime targets of many efforts in numerical simulation is the computation of the solutions for the convection-diffusion (CD) physical phenomena which are modeled by the CD equations. These latter are our model problem in this project.The convection-diffusion problems are of great practical importance since they arise in fluid flows, reactive flows, groundwater flows, traffic flows, two-phase flows in oil reservoirs etc. They also govern a variety of physical phenomena that appear in aeronautics, astrophysics, meteorology, semiconductors, financial modelling, and other areas. However, the numerical approximation of those problems is a major source of trouble, and no standard finite element can be used to approximate them. The weaknesses of the classical finite element methods to treat such problems, have pressed upon the attention to improve some methods and/or to develop new methods for more precise representative simulation.For these reasons, the objectives addressed here are to improve the popular Streamline-Upwind Petrov-G method and to develop new finite element methods, in order to avoid undesirable pathologies in the approximation of the solution of the CD equations. We apply these methods to perform numerical simulations of the general convection-diffusion physical phenomena, to the semiconductor, and to traffic flow problem, etc.