This book is the result of 25 years of study, reflection, and organization of ideas in the field of numerical discretization of PDEs while teaching courses in numerical analysis and searching for the best resource for teaching and learning the finite element (FE) method. The first part introduces the main building blocks of the FE discretization, together with analysis and algorithm development. The next part covers a specialized technique based on least squares approximation and multilevel preconditioning of variational formulations with different test and trial spaces. Various applications to elliptic boundary value problems, in particular, discretization to Stokes type systems and convection dominated problems, are included. Optimal norm on trial spaces, upwinding Petrov-Galerkin, and multilevel Uzawa algorithms are used for analysis and discretization. The last part focuses on special tools for FE analysis of PDEs on non-smooth domains, and PDEs with solutions in fractional spaces, such as the real method of interpolation, subspace interpolation, and Besov spaces. The organization of the ideas makes this book a freestanding textbook on FE including problem sets and algorithm implementation tips to support learning/advancing FE.
