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The finite element method is the most powerful general-purpose technique for computing accurate solutions to partial differential equations. This book contains a thorough derivation of the finite element equations as well as sections on programming the necessary calculations, solving the finite element equations, and using a posteriori error estimates to produce validated solutions. Accessible introductions to advanced topics, such as multigrid solvers, the hierarchical basis conjugate gradient method, and adaptive mesh generation, are provided. Each chapter ends with exercises to help readers master these topics. The book is practical in nature. It includes a carefully documented collection of MATLAB programs implementing the ideas presented. Readers will benefit from a detailed explanation of data structures and specific coding strategies and will learn how to write a finite element code from scratch. Students can use the MATLAB codes to experiment with the method and extend them in various ways.