Fitted Mesh Finite Element Method for Reaction-Diffusion Problems in One Dimension

Abstract

In this thesis, tted mesh nite element method has been presented for solving singularly perturbed one dimensional reaction-di usion equations. First, the given di erential equation is transformed to it's weak form and using shishkin mesh discretization technique the given domain is discretized in to a nite number of mesh elements so that piecewise linear base functions are de ned depending on this discretization. Then the approximate solution on each element is represented by taking the linear combination of the base functions. Substitution of the approximate solution to the weak form and applying Galerkin's method resulted a system of algebraic equations over each element. The obtained system of equations are then assembled to obtain the global system of equation and reduced to a nonsingular tridiagonal matrix which can be easily solved by inverse matrix method. To validate the applicability of the proposed method a model example is considered and solved for di erent values of the perturbation parameter and mesh elements. Numerical experiment is carried out to support the theoretical result using MATLAB software. The results have been presented in tables interms of maximum absolute error and graphs. The present method is "-uniform and approximates the exact solution very well.