09-23-2026, 08:15 PM
Mathematical Foundations of Finite Elements and Iterative Solvers by Paolo Gatto (SIAM, 2022) is an advanced textbook connecting the mathematical theory of the finite element method (FEM) with the numerical algorithms used to solve the resulting linear systems. It begins with partial differential equations arising in fluid dynamics and develops the functional-analytic foundations needed for FEM, including Sobolev spaces and elliptic problems. The book then explains Galerkin discretization, particularly continuous Galerkin methods, showing how infinite-dimensional PDE problems become finite-dimensional algebraic systems.
The second half focuses on the computational side: iterative solvers such as conjugate-gradient and GMRES methods, preconditioning, saddle-point systems, advection-diffusion equations, and the Stokes problem. A major theme is the interaction between mathematical analysis, discretization accuracy, and computational efficiency; for example, convection-dominated advection-diffusion problems can make standard Galerkin methods unstable and require stabilization techniques.
Overall, the 186-page book is aimed at graduate students and researchers in numerical analysis, applied mathematics, engineering, and scientific computing who want to understand not only how FEM works, but also the mathematical reasons behind its stability, convergence, and efficient numerical solution.
BOOK
The second half focuses on the computational side: iterative solvers such as conjugate-gradient and GMRES methods, preconditioning, saddle-point systems, advection-diffusion equations, and the Stokes problem. A major theme is the interaction between mathematical analysis, discretization accuracy, and computational efficiency; for example, convection-dominated advection-diffusion problems can make standard Galerkin methods unstable and require stabilization techniques.
Overall, the 186-page book is aimed at graduate students and researchers in numerical analysis, applied mathematics, engineering, and scientific computing who want to understand not only how FEM works, but also the mathematical reasons behind its stability, convergence, and efficient numerical solution.
BOOK
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