This book is the third volume of a three-part textbook suitable for graduate coursework
professional engineering and academic research. It is also appropriate for graduate flipped
classes. Each volume is divided into short chapters. Each chapter can be covered in one
teaching unit and includes exercises as well as solutions available from a dedicated website.
The salient ideas can be addressed during lecture with the rest of the content assigned as
reading material. To engage the reader the text combines examples basic ideas rigorous
proofs and pointers to the literature to enhance scientific literacy.Volume III is divided
into 28 chapters. The first eight chapters focus on the symmetric positive systems of
first-order PDEs called Friedrichs' systems. This part of the book presents a comprehensive and
unified treatment of various stabilization techniques from the existing literature. It
discusses applications to advection and advection-diffusion equations and various PDEs written
in mixed form such as Darcy and Stokes flows and Maxwell's equations. The remainder of Volume
III addresses time-dependent problems: parabolic equations (such as the heat equation)
evolution equations without coercivity (Stokes flows Friedrichs' systems) and nonlinear
hyperbolic equations (scalar conservation equations hyperbolic systems). It offers a fresh
perspective on the analysis of well-known time-stepping methods. The last five chapters discuss
the approximation of hyperbolic equations with finite elements. Here again a new perspective is
proposed. These chapters should convince the reader that finite elements offer a good
alternative to finite volumes to solve nonlinear conservation equations.
