Finite Elements II: Galerkin Approximation, Elliptic and Mixed Pdes

ISBN-13: 9783030569242 / Angielski / Miękka / 2022

Alexandre Ern;Jean-Luc Guermond

Opis

This book is the second 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 II is divided into 32 chapters plus one appendix. The first part of the volume 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. The material covered 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. Important implementation aspects regarding quadratures, linear algebra, and assembling are also covered. The remainder of Volume II focuses on PDEs where a coercivity property is available. It investigates conforming and nonconforming approximation techniques (Galerkin, boundary penalty, Crouzeix—Raviart, discontinuous Galerkin, hybrid high-order methods). These techniques are applied to 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.

Spis treści

Part V: Weak formulations and well-posedness.- Weak formulation of model problems.- Main results on well-posedness.- Part VI: Galerkin approximation.- Basic error analysis.- Error analysis with variational crimes.- Linear algebra.- Sparse matrices.- Quadratures.- Part VII: Elliptic PDEs: conforming approximation.- Scalar second-order elliptic PDEs.- H1-conforming approximation (I).- H1-conforming approximation (II).- A posteriori error analysis.- The Helmholtz problem.- Part VIII: Elliptic PDEs: nonconforming approximation.- Crouzeix-Raviart approximation.- Nitsche's boundary penalty method.- Discontinuous Galerkin.- Hybrid high-order methods.- Contrasted diffusivity (I).- Contrasted diffusivity (II).- Part IX: Vector-valued elliptic PDEs.- Linear elasticity.- Maxwell's equations: H(curl)-approximation.- Maxwell's equations: control on the divergence.- Maxwell's equations: further topics.- Part X: Eigenvalue problems.- Symmetric elliptic eigenvalue problems.- Symmetric operators, conforming approximation.- Nonsymmetric problems.- Part XI: PDEs in mixed form.- Well-posedness for PDEs in mixed form.- Mixed finite element approximation.- Darcy's equations.- Potential and flux recovery.- Stokes equations: Basic ideas.- Stokes equations: Stable Pairs (I).- Stokes equations: Stable pairs (II).- Appendices.- Bijective operators in Banach spaces.

O autorze

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.