*Finite Element Exterior Calculus* by Douglas N. Arnold is the first expository text covering the theory of finite element exterior calculus (FEEC), a developing area of research in the numerical analysis of partial differential equations (PDEs). As discussed below, FEEC furthers the study of the finite element method (FEM) approach to the numerical analysis of PDEs. This text is very well written with clear and detailed coverage of the theory of FEEC. After a short digression to provide context for FEEC, we review the presentation in *Finite Element Exterior Calculus*.

The numerical analysis of differential equations (DEs) is a major artery branching from the heart of applied mathematics. This area of applied mathematics remains active as a field of research and contributes heavily toward establishing a firm mathematical basis for computational science. This is especially true for the field of numerical analysis of PDEs. For an excellent overviews of the topics of differential equations (ordinary and partial), numerical analysis in general, and the numerical analysis of differential equations (ordinary and partial) specifically, I refer the reader to the obvious entries in the

*Princeton Companion to Applied Mathematics*.

Among the varying approaches to the numerical analysis of PDEs, the finite-element method (FEM) approach is widely appreciated for its power and beauty. Furthermore, recent software developments such as the open source

FEniCS project have dramatically increased the practical utility of finite-element methods (FEMs). With such tools, one can quickly formulate and run simulations for complex computational problems in applied partial differential equations in a very natural way that is also well suited for efficient computation.

It is often stated that the FEM approach to numerical PDEs has roots in the work of Richard Courant during the first half of the twentieth century. An interesting early treatment of finite-element approximation or at least principal aspects thereof for solutions to PDEs can be found in

*Approximate Methods of Higher Analysis* by Kantorovich and Krylov. Part of the appeal of FEMs is that they are supported upon an elegant mathematical foundation that closely parallels the linear functional analytic (or sometimes Hilbert space) approach to the mathematical theory of (weak) solutions of PDEs, techniques for which Courant's close colleague K. O. Friedrichs played a role in founding. See for example the article ``Partial Differential Equations in the 20th Century'' by Brezis and Browder. The idea here being to reformulate the problem of solving of a PDE as a problem of finding a solution to an equation involving a bilinear functional defined on a function space that (usually) strictly contains the space of differentiable functions, and often is a Hilbert space. The principal idea underlying the FEM approach to approximating a solution to a PDE is then to seek approximate solutions from a finite-dimensional subspaces of the function space over which the PDE problem was formulated in the aforementioned functional analytic framework, thereby producing a numerical method often referred to as a Galerkin method. The degree of accuracy of approximate solutions is typically a function of a discretization parameter(s) that determines the dimension of the finite-dimensional (Galerkin) subspace over which approximate solutions are sought. In practical settings one can then utilize the robust methods of numerical linear algebra to compute the approximate solutions. The functional analysis then also facilitates a reasonably direct if at times technical analysis of convergence and stability of the FEM numerical method. For a detailed coverage of the mathematical theory of FEMs see any of the books by Brenner and Scott, Ciarlet, Ern and Guermond, Johnson, or Oden and Reddy.

As Arnold states in his book, FEMs are ``mature tools in both practice and theory,'' yet ``there are still many important problems for which known numerical approaches fail.'' This is the motivation for the development of a newer theory, the finite element exterior calculus (FEEC), that both builds on and extends FEMs. The book *Finite Element Exterior Calculus* is Arnold's masterful explanation of FEEC for students of and researchers in the numerical analysis of PDEs.

The finite element exterior calculus was first defined as a coherent theory in a presentation by Arnold in 2002. Since then, a number of papers both by Arnold as well as other researchers have appeared that further develop the theory and practice of FEEC. The book *Finite Element Exterior Calculus*, based on a short course given by the author on FEEC in 2012, is meant to provide students and researchers interested in the numerical analysis of PDEs with a background in the mathematical apparatus of FEEC sufficient to enter into research in FEEC methods. Here the book succeeds admirably. A principal challenge of getting started with FEEC is that it is built on mathematical concepts that are not covered in typical courses in PDEs and numerical analysis for applied mathematicians, scientists, and engineers. Specifically, FEEC makes use of concepts from algebraic topology, the theory of smooth manifolds ( e.g. differential forms and their integration), and closed unbounded Hilbert space operators. The author develops these mathematical ideas in a clear and concise manner that is highly accessible to someone with training in applied mathematics, especially in the numerical analysis of PDEs.

A fundamental notion in the theoretical development of FEEC is the concept of a Hilbert complex. This is a (co)chain complex in the sense of homology (or homological algebra) that builds in additional functional analytic structure that is naturally associated with PDE problems. Following the definition given in *Finite Element Exterior Calculus*, a Hilbert complex is a sequence of Hilbert spaces \( W^{k} \) and a sequence of closed densely defined linear operators \( d^{k} \) from \( W^{k} \) to \( W^{k+1}\) such that \( \mbox{Range}(d^{k}) \subset \mbox{Kernel}(d^{k+1}) \). Further, associated with a Hilbert complex is a so-called Hodge Laplacian operator and much of the focus of FEEC is concerned with the numerical solution of PDEs arising from the Hodge Laplacian. An excellent presentation of all of the relevant background along with illustrative examples for the theory of Hilbert complexes and the Hodge Laplacian is provided in *Finite Element Exterior Calculus*.

The heart of *Finite Element Exterior Calculus* is concerned with developing an appropriate theory of weak solutions and a corresponding theory of Galerkin discretization for the Hodge Laplacian. This is all well-developed in Chapters 4 and 5 of the book. A running example throughout the book that both motivates and illustrates much of the theory is the de Rham complex, which arises naturally from the vector calculus operations of gradient, curl, and divergence applied to subsets of \( \mathbb{R}^{3} \). In order to generalize the theory to higher dimensions, the machinery of differential forms and their calculus is used. Arnold provides concise but thorough coverage of this material. The author then proceeds to develop a theory of finite element differential forms that provide a natural approach to obtaining the approximating (Galerkin) subspaces that play a role parallel to that of finite element spaces in the classical FEM theory. With these concepts and tools at hand, one is now prepared to delve further into the theory and application of FEEC. The books ends with some applications and further directions one might pursue in FEEC.

*Finite Element Exterior Calculus* provides an excellent path of entry into an area of cutting-edge research in applied mathematics. In the book, the author develops the foundations of finite element exterior calculus in a very clear manner and carefully motivates the theory with examples and applications. After reading this book, especially for those already versed in the classical theory of finite element methods, one is well-prepared to study the most recent research papers on FEEC and their applications. The book is an excellent resource for graduate students in applied mathematics focusing on numerical analysis and partial differential equations and I highly recommend it.

Jason M. Graham is an Associate Professor in the Department of Mathematics at the University of Scranton. He received a PhD in Applied Mathematical and Computational Sciences at the University of Iowa. Jason’s professional interests are in applied mathematics and mathematical biology.