Book excerpt: The finite element method is widely employed for numerical simulations in engineering and science due to its accuracy and efficiency. This concise introduction to the mathematical theory of the finite element method presents a selection of applications in civil and mechanical engineering including beams, elastic membranes, the wave equation, heat transfer, seepage in embankment, soil consolidation, incompressible fluids, and linear elasticity. Jupyter notebooks containing all Python programs of each chapter can be downloaded from the book's companion website. Arzhang Angoshtari is an assistant professor and Ali Gerami Matin is a graduate student, both in the department of Civil and Environmental Engineering at the George Washington University, USA. Their research interests cover theoretical and computational mechanics and finite element methods.