Monotone Discretizations for Elliptic Second Order Partial Differential Equations

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.

“The book is written with care and a clear didactic intention. Proofs are detailed, often at the level suitable for graduate students entering the field, while the breadth and depth of coverage will make it a standard reference for researchers working on discretisations for elliptic and convection-dominated problems. … Given the length and level of technicality, the book will be most beneficial to readers with a solid background in numerical analysis of PDEs and finite element theory.” (Denys Dutykh, Mathematical Reviews, May, 2026)

• - Introduction. - Convection-Di usion-Reaction Problems and Maximum Principles.- Discrete Maximum Principles.- Partitions of the Domain.- Finite Element Methods.- Finite Element Methods for Diffusion Problems.- Finite Element Methods for Reaction-Diffusion Problems.- Linear Finite Element Methods for Convection-Diffusion-Reaction Problems.- Nonlinear Finite Element Methods for Convection-Diffusion-Reaction Problems: Discretizations Based on Modi ed Variational Forms.- Nonlinear Finite Element Methods for Convection-Diffusion-Reaction Problems: Algebraically Stabilized Methods.- Finite Difference Methods.- Finite Volume Methods.- A Numerical Study for a Problem with Different Regimes.- Outlook.