Theory and Practice of Finite ElementsSpringer Science & Business Media, 09.03.2013 - 526 Seiten The origins of the finite element method can be traced back to the 1950s when engineers started to solve numerically structural mechanics problems in aeronautics. Since then, the field of applications has widened steadily and nowadays encompasses nonlinear solid mechanics, fluid/structure interactions, flows in industrial or geophysical settings, multicomponent reactive turbulent flows, mass transfer in porous media, viscoelastic flows in medical sciences, electromagnetism, wave scattering problems, and option pricing (to cite a few examples). Numerous commercial and academic codes based on the finite element method have been developed over the years. The method has been so successful to solve Partial Differential Equations (PDEs) that the term "Finite Element Method" nowadays refers not only to the mere interpolation technique it is, but also to a fuzzy set of PDEs and approximation techniques. The efficiency of the finite element method relies on two distinct ingredi ents: the interpolation capability of finite elements (referred to as the approx imability property in this book) and the ability of the user to approximate his model (mostly a set of PDEs) in a proper mathematical setting (thus guar anteeing continuity, stability, and consistency properties). Experience shows that failure to produce an approximate solution with an acceptable accuracy is almost invariably linked to departure from the mathematical foundations. Typical examples include non-physical oscillations, spurious modes, and lock ing effects. In most cases, a remedy can be designed if the mathematical framework is properly set up. |
Inhalt
3 | |
4 | |
Approximation in Banach Spaces by Galerkin Methods | 81 |
Coercive Problems 111 | 110 |
Mixed Problems | 175 |
FirstOrder PDEs | 219 |
TimeDependent Problems | 279 |
Data Structuring and Mesh Generation 337 | 336 |
Linear Algebra | 383 |
A Posteriori Error Estimates and Adaptive Meshes | 421 |
Syllabus 1 1 to 1 5 Chapters 24 5 1 5 4 6 2 Chapters 7 and 9 | 450 |
A Banach and Hilbert Spaces 463 | 462 |
B Functional Analysis | 477 |
Nomenclature | 493 |
Acknowledgments We are indebted to many colleagues and former stu | 501 |
Author Index 513 | 512 |
Andere Ausgaben - Alle anzeigen
Theory and Practice of Finite Elements Alexandre Ern,Jean-Luc Guermond Eingeschränkte Leseprobe - 2004 |
Theory and Practice of Finite Elements Alexandre Ern,Jean-Luc Guermond Keine Leseprobe verfügbar - 2013 |
Theory and Practice of Finite Elements Alexandre Ern,Jean-Luc Guermond Keine Leseprobe verfügbar - 2010 |
Häufige Begriffe und Wortgruppen
affine algorithm approximation space Assume Banach space barycentric coordinates bijective bilinear form boundary conditions bounded Bowyer-Watson algorithm coercivity compute condition number Consider continuous embedding convergence Corollary defined Definition degrees of freedom denote dimension Dirichlet problem domain edges equation error estimates Example Exercise finite element method first-order Galerkin Galerkin methods geometric global shape functions H¹-conformal H¹(N Hilbert space implies independent of h inf-sup condition integral interpolation operator introduce iterative Ker(B L²(N Lagrange finite element Lagrange polynomials Lax-Milgram Lemma Lemma linear system meas(K mesh nodes norm Owing P₁ PDEs polynomials Proof Proposition Prove quadrature Remark result satisfies scalar Seek solution solve stability Stokes problem surjective symmetric technique triangle velocity weak formulation well-posed yields