Finite Element Analysis of a One-dimensional Helmholtz Equation

Many science and engineering problems can be mathematically modeled as a boundary value problem. Thus, it is very important to build a method to solve for a boundary value problem. For some simple boundary value problems, for example, a two-dimensional Laplace equation, it may be possible to solve it analytically by separation of variables. However, in most applications, boundary value problems are much more complex and there are no available analytical methods. For this reason, a numerical method called finite element method (FEM) is developed for the analysis of boundary value problems. It approximates the domain by many small elements and simplify possible integrations. In this paper, FEM is applied to solve a boundary value problem of a one-dimensional Helmholtz equation. It can be seen that FEM provides a fast and accurate approach for this problem.