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.

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Gabriel R. Barrenechea obtained his Mathematical Engineering degree at the Universidad de Concepción, Chile, in 1997. He then obtained his Doctorate in Sciences from Université Paris Dauphine, France, in 2002. The same year he became Assistant (then, Associate) Professor at the Universidad de Concepcion, where he worked until 2007. He then moved to the University of Strathclyde, where he is a Reader in Numerical Analysis. His main field of research is the development and mathematical analysis of new finite element methods, especially for problems in incompressible fluid mechanics (Newtonian and non-Newtonian), with an emphasis on physical consistency of the methods. He has edited two invited volumes, and (co-)authored over 50 scientific publications in refereed scientific journals.

Volker John studied mathematics in Halle (1992). He obtained his Ph.D. degree 1997 in Magdeburg, where he also wrote his habilitation thesis (2002). In 2004 he became professor for 'Applied Mathematics' at the Saarland University in Saarbrücken. Since 2009 he has been head of the research group

'Numerical Mathematics and Scientific Computing' at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin and he has been professor for 'Numerics of Partial Differential Equations' at the Freie Universität Berlin. His main fields of research are finite element methods for scalar convection-diffusion equations and for incompressible flow problems. He is interested as well in the numerical analysis of these methods as in using them in applications. He is author of two monographs and (co-)authored more than 100 papers in refereed scientific journals.

Petr Knobloch studied mathematics and physics at the Charles University in Prague (1993) and obtained his Ph.D. degree from the Otto-von-Guericke Universität Magdeburg in 1996. After a one-year postdoc position in Magdeburg, he moved back to the Charles University where he habilitated in 2006. Since then he was associate professor in Numerical Mathematics. In 2017 he was awarded the scientific title Research Professor in Physico-Mathematical Sciences from the Czech Academy of Sciences. In 2024 he became full professor at the Charles University in Prague. His scientific interests cover various aspects of the finite element method but the main emphasis has been on the numerical solution of singularly perturbed problems. He has edited two invited volumes, and (co-)authored over 50 papers in refereed scientific journals.