Abstract:
Abstract
This thesis treats the heat equation and its approximate solution using the
nite element method. For this purpose, a general theory of the nite ele-
ment method and its strategy are explained carefully. Then after, the heat
equation and its derivation and applications are considered in details. The
nite element method is then applied to approximate the solution of the heat
equation in one and two dimensional spaces. In this essence, error analysis is
studied for the heat equation in both categories, a posteriori and a priori error
estimates.
Also, the thesis discusses the numerical solution for the heat equation through-
out examples in one and two dimensions using the MATLAB software. Since
the heat equation requires time discretization besides the spatial one, then
iterative methods are needed for the time solution. In one dimension, we ap-
ply two methods for the time iteration, the backward Euler method and theta
method. It is found that Theta method in the FEM is more stable than the
Backward Euler method: increasing number of nodal points provides better
convergence to the exact solution, i.e., less error. Been studying of , sev-
eral values of are tested to reach the optimal value which makes the error
as minimum as possible. After examining several choices of , we arrive at
the conclusion that should live in the interval [0:65; 0:8] to obtain better
convergence.
