Solving an Advection-Diffusion Equation by a Finite Element Method

Abstract

In this project we seek a numerical approximation of the solution u : [0, 1] → ℝ of the following problem:

$$ \left\{ \begin{gathered} - \varepsilon u''\left( x \right) + \lambda u'\left( x \right) = f\left( x \right), x \in \left] {0,1} \right[, \hfill \\ u\left( 0 \right) = 0, \hfill \\ u\left( 1 \right) = 0. \hfill \\ \end{gathered} \right. $$

((4.1))

The function f and the real numbers ε > 0 and λ are given in such a way that there exists a unique continuous solution of this problem. Our aim is to approximate the solution with a continuous piecewise polynomial function.