Monatshafte fur Math 126:3, 1998

Reviewer: H. Muthsam

This text on computational differential equations, which ultimately centers on the Galerkin method for linear partial differential equations, starts at a quite elementary mathematical level, namely by reviewing calculus and basic linear algebra. In doing so, much of the material proper to be described later on is preshaped. For example, when dealing with the fundamental theorem of calculus, the idea of adaptivity (in numerical quadrature) is advanced. The Galerkin method, then, is also introduced in the most elementary setting, namely an initial value problem for an ordinary differential equation. Here, ill condition pops up for the first time. The solution of linear systems of algebraic equations is then pursued with more vigour. Subsequently, initial and boundary value problems are investigated in one and several dimensions within the Galerkin framework for the most important type of linear equations. Much attention is given to both error estimates and adaptivity, the latter including foremost the generation of grids. - Due to the book's extensive discussion, it will also be useful for the scientist or engineer who needs to employ the methods.