Ciarlet, P. G. (ed.) et al., Handbook of numerical analysis. Vol. 7: Solution of equations in $\Bbb R^n$ (Part 3). Techniques of scientific computing (Part 3). Amsterdam: North-Holland/ Elsevier, 713-1020 (2000).

The (part of) book under review is a survey of finite volume methods. It begins with a concise introduction. Basic features of finite volume methods are outlined on two typical examples of partial differential equations (PDEs), namely the Laplace equation (of elliptic type) and the 2-dimensional transport equation (of hyperbolic type). Comparison is made with other discretization techniques (finite difference methods and finite element methods), and a number of bibliographic references is given. In subsequent chapters, finite volume methods are presented and studied in more details for various types of PDEs. The exposition is made clear by always going from the simplest cases to more complicated ones, and is enriched by numerous remarks addressing further extensions.
One chapter is devoted to one dimensional elliptic PDEs. The finite volume method is first applied to the Laplace equation ($-u''=f$) under Dirichlet boundary conditions. It is compared in that framework to the mixed finite element method. An ${\cal O}(h)$ error estimate is derived for (even unstructured) meshes of size $h$, and convergence is proved under a regularity assumption on the exact solution. The authors then turn to more general (linear) elliptic PDEs, and finally prove a convergence result for the semi-linear equation $$-u''=f(x,u)$$ under Dirichlet boundary conditions. In the latter case, very few regularity is required for $f$, and the solution obtained by passing to the limit in the numerical scheme is a weak solution.
The next chapter extends the approach to 2-dimensional and 3-dimensional elliptic equations. The equation of interest takes the form $$-\Delta u+\text{div}({\bold v} u)+bu=f$$ with the vector field ${\bold v}$ satisfying $\text{div}{\bold v}\geq 0$. The authors first show (in the case ${\bold v}=0$, $b=0$) that for rectangular meshes the scheme is an easy extension of the one-dimensional case, thus proving an ${\cal O}(h)$ error estimate. More general meshes are then considered. A crucial notion of admissible mesh is introduced (an example of which is the so-called Voronoi mesh). Convergence theorems are then proved under Dirichlet conditions, homogeneous or not, and under Neumann conditions. They all rely on discrete Poincaré inequalities and strong compactness arguments. At the end of the chapter, more general elliptic operators (with matrix valued diffusions) and more general boundary conditions are also considered.
The authors then extend their techniques to evolution equations of parabolic type, $$\partial_tu-\Delta \varphi(u)+\text{div}({\bold v} u)+bu=f.$$ They prove an error estimate in the linear case ($\varphi=$id) under Dirichlet boundary conditions. It is in ${\cal O}(h+k)$ if $h$ is the space mesh size and $k$ is the time step. They also prove convergence in the case ${\bold v}=0$, $b=0$ with homogeneous Neumann conditions.
Another field of application of finite volume methods is the hyperbolic conservation laws. One chapter is devoted to one-dimensional equations. It begins with a careful discussion of entropy solutions and Krushkov’s theorem, on the illuminating example of the $N$-wave. The upwind finite volume scheme is introduced for the linear equation $\partial_tu + \partial_x u = 0$, and compared to the upwind finite difference scheme. Besides the classical $L^\infty$ stability estimate, the authors show another crucial estimate, named weak BV estimate, of which they give an interpretation in terms of the second order equivalent equation (of parabolic type). For nonlinear equations of the kind $$\partial_tu + \partial_x f(u) = 0,$$ finite volume schemes are based on so-called numerical fluxes (supposed to approximate $f(u)$). The notion of numerical flux is also used in conservative finite difference schemes, but it plays an even more crucial role here. The authors recall the definitions of consistency and monotony of numerical fluxes. They prove that monotone schemes are $L^\infty$ stable, satisfy discrete entropy inequalities and weak BV estimates, which in turn implies convergence using the concept of entropy process solution. In passing, the well-known theorem of Lax-Wendroff is recalled and proved. Higher order (MUSCL) schemes are briefly outlined.
After that introductory chapter, the approximation of multidimensional hyperbolic equations is considered. The exposition concerns equations of the form $$\partial_t u +\text{div}({\bold v} f(u)) =0$$ where ${\bold v}$ is a divergence free vector field. Such equations occur in porous medium modelling, generally coupled with elliptic or parabolic equations. Like in the one-dimensional case, the ingredients for convergence are $L^\infty$ stability, discrete entropy inequalities and weak BV estimates. They are proved for both the explicit and implicit schemes (of which solutions are shown to exist by means of topogical degree). Furthermore, for BV initial data, an ${\cal O}(h^{1/4})$ error estimate is proved.
The last chapter is the only one dealing with systems of PDEs, for which both mathematical and numerical analysis are mainly open. Various kinds of systems are considered. For hyperbolic systems, the Godunov scheme and the Roe scheme, together with variants like the one called VFRoe, are reviewed. The question of boundary conditions is briefly addressed. Another part concerns the stationary incompressible Navier-Stokes equation, and the last one is about models arising in porous media. For the entire collection see [

Zbl 0953.00016].