Beschreibung
This book is the second volume of a three-part textbook suitable for both graduate coursework and professional engineering and research reference alike. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters that can be covered in one teaching unit and includes exercises as well as solutions available from a dedicated website. The structure allows for the salient ideas to be addressed during lecture, with the rest of the content assigned as reading material. To engage the reader, the text's exposition combines motivating examples, basic ideas, rigorous proofs, and pointers to the literature to enhance scientific literacy. Volume II is divided into 32 chapters plus one appendix. It focuses on the approximation of elliptic and mixed PDEs, beginning with fundamental results on well-posed weak formulations and their approximation by the Galerkin method. This section includes key results such as the BNB theorem based on inf-sup conditions, Céa's and Strang's lemmas, and the duality argument by Aubin and Nitsche. It also addresses important implementation aspects regarding quadratures, linear algebra, and assembling. The remainder of Volume II focuses on PDEs where a coercivity property is available, such as conforming and nonconforming approximation techniques (Galerkin, boundary penalty, Crouzeix-Raviart, discontinuous Galerkin, hybrid high-order methods). In addition, this volume considers applications of elliptic PDEs (diffusion, elasticity, the Helmholtz problem, Maxwell's equations), eigenvalue problems for elliptic PDEs, and PDEs in mixed form (Darcy and Stokes flows). Finally, the appendix addresses fundamental results on the surjectivity, bijectivity, and coercivity of linear operators in Banach spaces.
Autorenporträt
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. JeanLuc 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 coauthored two books and over 170 research papers in peerreviewed journals.
