This book provides an accessible yet rigorous introduction to the mathematical analysis of finite element methods (FEMs), which serve as powerful computational tools supported by solid mathematical foundations. FEMs are built on weak or variational formulations of differential equations and on piecewise polynomial approximations, making them particularly effective for boundary value problems of elliptic partial differential equations.While many books address the mathematical theory of FEMs, they are often written at a level that is difficult for beginners, especially students in engineering. This book aims to bridge that gap by presenting the most essential and fundamental aspects of FEM analysis in a concise and self-contained manner without compromising mathematical depth. To maintain clarity and focus, the discussion is limited to one- and two-dimensional differential equations, and the finite elements considered are among the simplest. The bibliography is deliberately selective, referring mainly to classical and representative works.The first part of the book introduces the mathematical foundations of FEMs through the analysis of the two-dimensional Poisson equation. These chapters are suitable for undergraduate students and are designed to provide a clear overview of the subject. Chap. 1 offers a short introductory course from a strict mathematical viewpoint, giving readers a broad perspective on the field.The subsequent chapters develop weak formulations of 2D Poisson boundary value problems, their simplest finite element approximations, and corresponding error estimates. In the later part of the volume, the focus shifts to saddle-point type approximation problems, which play important roles in areas such as fluid mechanics, solid mechanics, and electromagnetism. Topics on hypercircle method and eigenvalue estimation are also included to enrich the reader's understanding.Finally, the appendices present essential notations and theorems used throughout the book, often in their simplest and most illustrative forms. This structure enables beginners to approach FEMs with confidence while still offering specialists a mathematically sound and meaningful treatment.
