Volker John – författare

This book offers a comprehensive presentation of numerical methods for elliptic boundary value problems that satisfy discrete maximum principles (DMPs). The satisfaction of DMPs ensures that numerical solutions possess physically admissible values, which is of utmost importance in numerous applications. A general framework for the proofs of monotonicity and discrete maximum principles is developed for both linear and nonlinear discretizations. Starting with the Poisson problem, the focus is on convection-diffusion-reaction problems with dominant convection, a situation which leads to a numerical problem with multi-scale character. The emphasis of this book is on finite element methods, where classical (usually linear) and modern nonlinear discretizations are presented in a unified way. In addition, popular finite difference and finite volume methods are discussed. Besides DMPs, other important properties of the methods, like convergence, are studied. Proofs are presented step by step, allowing readers to understand the analytic techniques more easily. Numerical examples illustrate the behavior of the methods.

This book explores finite element methods for incompressible flow problems: Stokes equations, stationary Navier-Stokes equations and time-dependent Navier-Stokes equations. It focuses on numerical analysis, but also discusses the practical use of these methods and includes numerical illustrations. It also provides a comprehensive overview of analytical results for turbulence models. The proofs are presented step by step, allowing readers to more easily understand the analytical techniques.