The present work focuses on using and applying spectral element methods for investigating flow characteristics. Most numerical studies for analyzing fluid flow characteristics were carried out using the three traditional lower-order methods: finite difference, finite volume, and finite element method. The main demerits of these methods are computational cost and accuracy. The spectral element method is an extension of the finite element method which uses higher-order polynomials like the spectral methods, thus having a higher-order accuracy with less computational time and complex geometry handling capacity. A comparison of the in-house solver with commercially available software has been done. A numerical simulation of flow past a stationary and sinusoidally oscillating square cylinder at Re = 100 and 150 was conducted and compared with previous studies to check the correctness of the solver. The main objective is to develop an efficient solver to simulate flow-induced vibration problems.