Finite Elements III

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.

Short chapters allow for development of ideas in a classroom setting Many exercises and hints included Well adapted for graduate flipped classes Part of a three volume work

Autorentext
Alexandre Ern is Senior Researcher at Ecole des Ponts and INRIA in Paris, and he is also Associate Professor of Numerical Analysis at Ecole Polytechnique, Paris. His research deals with the devising and analysis of finite element methods and a posteriori error estimates and adaptivity with applications to fluid and solid mechanics and porous media flows. Alexandre Ern has co-authored three books and over 150 papers in peerreviewed journals. He has supervised about 20 PhD students and 10 postdoctoral fellows, and he has ongoing collaborations with several industrial partners. Jean-Luc Guermond is Professor of Mathematics at Texas A&M University where he also holds an Exxon Mobile Chair in Computational Science. His current research interests are in numerical analysis, applied mathematics, and scientific computing. He has co-authored two books and over 170 research papers in peer-reviewed journals.

Inhalt
Part XII: First-order PDEs.- Friedrichs' systems.- Residual-based stabilization.- Fluctuation-based stabilization (I).- Fluctuation-based stabilization (II).- Discontinuous Galerkin.- Advection-diffusion.- Stokes equations: Residual-based stabilization-. Stokes equations: other stabilizations.- Part XIII: Parabolic PDEs.- Bochner integration.- Weak integration and well-posedness.- Semi-discretization in space.- Implicit and explicit Euler schemes.- BDF2 and Crank-Nicolson schemes.- Discontinuous Galerkin in time.- Continuous Petrov-Galerkin in time.- Analysis using inf-sup stability.- Part XIV: Time-dependent Stokes equations.- Weak formulations and well-posedness.- Monolithic time discretization.- Projection methods.- Artificial compressibility.- Part XV: Time-dependent first-order linear PDEs.- Well-posedness and space semi-discretization.- Implicit time discretization.- Explicit time discretization.- Part XVI: Nonlinear hyperbolic PDEs.- Scalar conservation equations.- Hyperbolic systems.- First-order approximation.- Higher-order approximation.- Higher-order approximation and limiting.