Galerkin methods for one-dimensional advection diffusion equation

Abstract

We consider the finite element Galerkin method to solve the one-dimensional advection diffusion equation with homogeneous Dirichlet conditions. We start by introducing the method for the steady heat equation and then we present the Petrov-Galerkin modifications associated with the addition of an advection term. As last step, we discuss some techniques used to deal with the introduction of a time-dependent derivative. An in-house code is written in C++ to model the equations leading to the full advection diffusion equation. An example is considered for each one of the 3 cases and the numerical solution obtained from the code is plotted along with the exact solution. The post processing is done in Matlab.