Mathematical Theory of Finite Elements by Leszek F. Demkowicz (SIAM, 2023) is a graduate-level treatment of the mathematical foundations behind the finite element method (FEM), with particular emphasis on stability, variational formulations, and convergence. The book begins with the calculus of variations, weak formulations, Hilbert spaces, and the Galerkin method, then develops the theory of coe$rcive problems through the Ritz method, the Lax–Milgram theorem, and Céa’s lemma. It subsequently explains how conforming finite elements are constructed within the exact grad–curl–div sequence, covering spaces such as $H1H^1$, $H(curl)H(\mathrm{curl})$, $H(div)H(\mathrm{div})$, and $L2L^2$.
The second half moves beyond coercive problems to the more difficult situations encountered in wave propagation, electromagnetics, and mixed formulations. Demkowicz develops Babuška’s discrete inf-sup condition, Mikhlin’s theory, Brezzi’s theory for mixed methods, and the role of adaptive mesh refinement in recovering optimal convergence for singular solutions.
The final chapter introduces the Discontinuous Petrov–Galerkin (DPG) method, including optimal test functions, broken test spaces, Fortin operators, and double adaptivity. Overall, the book is less a practical FEM programming manual than a rigorous mathematical framework explaining why finite element methods are stable and converge, making it particularly relevant to graduate students and researchers in numerical analysis, PDEs, computational mathematics, and engineering.
BOOK
The second half moves beyond coercive problems to the more difficult situations encountered in wave propagation, electromagnetics, and mixed formulations. Demkowicz develops Babuška’s discrete inf-sup condition, Mikhlin’s theory, Brezzi’s theory for mixed methods, and the role of adaptive mesh refinement in recovering optimal convergence for singular solutions.
The final chapter introduces the Discontinuous Petrov–Galerkin (DPG) method, including optimal test functions, broken test spaces, Fortin operators, and double adaptivity. Overall, the book is less a practical FEM programming manual than a rigorous mathematical framework explaining why finite element methods are stable and converge, making it particularly relevant to graduate students and researchers in numerical analysis, PDEs, computational mathematics, and engineering.
BOOK
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