Approximation of Convection-Dominated Problems

• Olivia Karney, Applied Mathematics, University of Delaware
• Scott Bria, Applied Mathematics, University of Delaware

• Constantin Bacuta, Mathematical Sciences, University of Delaware

We will present basic finite difference and finite element discretizations in one dimension for a convection-dominated problem. This is a Convection-Diffusion boundary value problem, which depends on a diffusion parameter, epsilon, and exhibits sharp fluctuations in solutions (boundary layers). The challenge is to approximate the exact solution using the finite difference method and the finite element method when the discretization parameter, h, is much larger than the diffusion parameter, epsilon. We estimated the discrete matrices for convection and diffusion for both methods. Without upwinding, both approximations displayed nonphysical oscillations. To minimize these oscillations, we used the upwinding technique for both methods. Upwinding allows us to obtain well-conditioned matrices. We implemented these two methods in MATLAB and showed numerical results to compare the two methods. After utilizing upwinding with both the finite difference and finite element methods, we discovered that for the same values of epsilon and h, the finite element method consistently provided results that were more accurate when compared to the exact solution. We experimented with various upwinding functions as well as different forcing functions to conclude that the finite element method provides a better numerical solution of our Convection-Diffusion problem.  To improve our results with the finite element method, we experimented with different quadratures to approximate the dual vector and  obtain solutions that better approximate the exact solution.