Numerical Solution of the Heat Equation Using Finite Element Method

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.