Finite Element Approximations of Burgers' Equation with Robin's Boundary Conditions

Abstract

This work is a numerical study of Burgers' equation with Robin's boundary conditions. The goal is to determine the behavior of the solutions in the limiting cases of Dirichlet and Neumann boundary conditions. We develop and test two separate finite element and Galerkin schemes. The Galerkin/Conservation method is shown to give better results and is then used to compute the response as the Robin's Boundary conditions approach both the Dirichlet and Neumann boundary conditions. Burgers' equation is treated as a perturbation of the linear heat equation with the appropriate realistic constants. The goal is to determine if the use of the Robin's boundary conditions to approximate Dirichlet and Neumann boundary conditions affords any advantage over schemes that employ only "exact" Dirichlet or Neumann boundary conditions. Our finite element results indicate that solutions using appropriate Robin's boundary conditions approach the same solutions obtained by "exact" Dirichlet or Neumann boundary conditions. This allows us to obtain realistic solutions in some cases where the other schemes had previously failed.