This book provides the theoretical foundation necessary to develop efficient methods for solving partial differential equations. In addition to the standard convergence theory, it is the first text of its kind to address other ingredients required for successful numerical simulations of physical systems. The presentation is accessible to scientists and engineers with diverse backgrounds and interests, ranging from application modeling to numerical analysis and scientific software development. The content is strongly influenced by the authors' research in space physics, electrical and optical engineering, applied mathematics, numerical analysis and professional software development.