This comprehensive book focuses on numerical methods for approximating solutions to partial differential equations. Intended as a broad survey of methods, the aim is to introduce readers to the central concepts of various families of discretizations and solution algorithm and lay the foundation needed to understand more advanced material, The book is divided into four parts: Part I covers basic background on PDEs and numerical methods,Part II introduces the three main classes of numerical methods for PDEs that are the book’s focus (finite-difference, finite-element, and finite-volume methods),Part III discusses linear solvers and finite-element and finite-volume methods at a more advanced level, andPart IV presents further high-level topics on discretizations and solvers.The authors include over one hundred well-established definitions, theorems, corollaries, and lemmas, and summaries of and reference to in-depth treatments of more advanced mathematics when needed.Audience This book is intended for advanced undergraduate/first-year graduate and advanced graduate students in applied math, as well as students in science and engineering disciplines. It is also appropriate for researchers in the field of scientific computing.Chapters are designed to be stand-alone, allowing distinct paths through the text, making it appropriate for both single-semester and multisemester courses. It is appropriate for courses on numerical methods for PDEs and numerical linear algebra.