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This book presents the mathematical theory of finite elements, starting from basic results on approximation theory and finite element interpolation and building up to more recent research topics, such as and Discontinuous Galerkin, subgrid viscosity stabilization, and a posteriori error estimation. The body of the text is organized into three parts plus two appendices collecting the functional analysis results used in the book. The first part develops the theoretical basis for the finite element method and emphasizes the fundamental role of inf-sup conditions. The second party addresses various applications encompassing elliptic PDE's, mixed formulations, first-order PDEs, and the time-dependent versions of these problems. The third part covers implementation issues and should provide readers with most of the practical details needed to write or understand a finite element code. Written at the graduate level, the text contains numerous examples and exercises and is intended to serve as a graduate textbook. Depending on one's interests, several reading paths can be followed, emphasizing either theoretical results, numerical algorithms, code efficiency, or applications in the engineering sciences. The book will be useful to researchers and graduate students in mathematics, computer science and engineering.
Finite element method. --- Méthode des éléments finis --- Finite element method --- Méthode des éléments finis --- Mathematical analysis. --- Analysis (Mathematics). --- Applied mathematics. --- Engineering mathematics. --- Computer science—Mathematics. --- Partial differential equations. --- Computer mathematics. --- Analysis. --- Applications of Mathematics. --- Math Applications in Computer Science. --- Partial Differential Equations. --- Computational Mathematics and Numerical Analysis. --- Mathematical and Computational Engineering. --- Computer mathematics --- Electronic data processing --- Mathematics --- Partial differential equations --- Engineering --- Engineering analysis --- Mathematical analysis --- 517.1 Mathematical analysis --- Éléments finis, Méthode des --- Acqui 2006 --- Éléments finis, Méthode des.
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Calculus. --- Functional analysis. --- Functions. --- Harmonic analysis. --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Differential equations --- Numbers, Complex --- Set theory --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Functions --- Geometry, Infinitesimal --- Mètode dels elements finits --- Equacions en derivades parcials --- EDPs --- Equació diferencial en derivades parcials --- Equacions diferencials en derivades parcials --- Equacions diferencials parcials --- Equacions diferencials --- Dispersió (Matemàtica) --- Equació d'ona --- Equació de Dirac --- Equació de Fokker-Planck --- Equació de Schrödinger --- Equacions de Navier-Stokes --- Equacions de Hamilton-Jacobi --- Equacions de Maxwell --- Equacions de Monge-Ampère --- Equacions de Von Karman --- Equacions diferencials el·líptiques --- Equacions diferencials hiperbòliques --- Equacions diferencials parabòliques --- Equacions diferencials parcials estocàstiques --- Funcions harmòniques --- Laplacià --- Problema de Cauchy --- Problema de Neumann --- Teoria espectral (Matemàtica) --- Anàlisi numèrica --- Equacions de Von Kármán
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Calculus. --- Functional analysis. --- Functions. --- Harmonic analysis. --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Differential equations --- Numbers, Complex --- Set theory --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Functions --- Geometry, Infinitesimal --- Mètode dels elements finits --- Equacions en derivades parcials --- Anàlisi funcional --- Mètodes de Galerkin --- Mètodes de Sinc --- Mètodes de Sinc-Galerkin --- Anàlisi numèrica --- Càlcul funcional --- Càlcul de variacions --- Àlgebres de Hilbert --- Àlgebres topològiques --- Anàlisi funcional no lineal --- Anàlisi microlocal --- Espais analítics --- Espais de Hardy --- Espais d'Orlicz --- Espais funcionals --- Espais vectorials normats --- Espais vectorials --- Filtres digitals (Matemàtica) --- Funcionals --- Funcions vectorials --- Multiplicadors (Anàlisi matemàtica) --- Pertorbació (Matemàtica) --- Teoria d'operadors --- Teoria de distribucions (Anàlisi funcional) --- Teoria de functors --- Teoria de l'aproximació --- Teoria del funcional de densitat --- Teoria espectral (Matemàtica) --- Equacions funcionals --- Equacions integrals --- EDPs --- Equació diferencial en derivades parcials --- Equacions diferencials en derivades parcials --- Equacions diferencials parcials --- Equacions diferencials --- Dispersió (Matemàtica) --- Equació d'ona --- Equació de Dirac --- Equació de Fokker-Planck --- Equació de Schrödinger --- Equacions de Navier-Stokes --- Equacions de Hamilton-Jacobi --- Equacions de Maxwell --- Equacions de Monge-Ampère --- Equacions de Von Karman --- Equacions diferencials el·líptiques --- Equacions diferencials hiperbòliques --- Equacions diferencials parabòliques --- Equacions diferencials parcials estocàstiques --- Funcions harmòniques --- Laplacià --- Problema de Cauchy --- Problema de Neumann --- Equacions de Von Kármán
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This book is the first 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 I is divided into 23 chapters plus two appendices on Banach and Hilbert spaces and on differential calculus. This volume focuses on the fundamental ideas regarding the construction of finite elements and their approximation properties. It addresses the all-purpose Lagrange finite elements, but also vector-valued finite elements that are crucial to approximate the divergence and the curl operators. In addition, it also presents and analyzes quasi-interpolation operators and local commuting projections. The volume starts with four chapters on functional analysis, which are packed with examples and counterexamples to familiarize the reader with the basic facts on Lebesgue integration and weak derivatives. Volume I also reviews important implementation aspects when either developing or using a finite element toolbox, including the orientation of meshes and the enumeration of the degrees of freedom.
Functional analysis --- Differential equations --- differentiaalvergelijkingen --- functies (wiskunde)
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This book introduces the basic ideas for building discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. It is to a large extent self-contained and is intended for graduate students and researchers in numerical analysis. The material covers a wide range of model problems, both steady and unsteady, elaborating from advection-reaction and diffusion problems up to the Navier-Stokes equations and Friedrichs' systems. Both finite-element and finite-volume viewpoints are utilized to convey the main ideas underlying the design of the approximation. The analysis is presented in a rigorous mathematical setting where discrete counterparts of the key properties of the continuous problem are identified. The framework encompasses fairly general meshes regarding element shapes and hanging nodes. Salient implementation issues are also addressed.
Discontinuous functions. --- Engineering mathematics. --- Galerkin methods. --- Galerkin methods --- Discontinuous functions --- Engineering mathematics --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Calculus --- Applied Mathematics --- Engineering --- Engineering analysis --- Functions, Discontinuous --- Sinc-Galerkin methods --- Sinc methods --- Mathematics. --- Computer mathematics. --- Numerical analysis. --- Applied mathematics. --- Numerical Analysis. --- Computational Mathematics and Numerical Analysis. --- Appl.Mathematics/Computational Methods of Engineering. --- Mathematical analysis --- Functions --- Numerical analysis --- Computer science --- Mathematical and Computational Engineering. --- Computer mathematics --- Discrete mathematics --- Electronic data processing
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With the increasing awareness of the heavy burden placed on environmental resources and the need for industry and public institutions to cope with more stringent regulations, this timely book focuses on some specific, but very important, environmental problems, namely, surface and subsurface hydrosystems. Covering state-of-the-art techniques to model such systems, the volume will be of great benefit to all researchers in applied mathematics and environmental engineering.
Hydrology --- Physical geography. --- Geography --- Mathematical models.
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Elasticitat --- Models matemàtics --- Física de l'estat sòlid --- Física --- Dinàmica reticular --- Electrònica de l'estat sòlid --- Estructura cristal·lina (Sòlids) --- Micromecànica --- Vidres de spin --- Sòlids --- Models (Matemàtica) --- Models experimentals --- Models teòrics --- Mètodes de simulació --- Anàlisi de sistemes --- Mètode de Montecarlo --- Modelització multiescala --- Models economètrics --- Models lineals (Estadística) --- Models multinivell (Estadística) --- Models no lineals (Estadística) --- Programació (Ordinadors) --- Simulació per ordinador --- Teoria de màquines --- Models biològics --- Estàtica --- Física matemàtica --- Impacte --- Mecànica --- Mecànica analítica --- Propietats de la matèria --- Aeroelasticitat --- Equacions de Von Karman --- Fotoelasticitat --- Histèresi --- Mecànica dels medis continus --- Ones elàstiques --- Termoelasticitat --- Plasticitat --- Viscoelasticitat --- Esforç i tensió --- Reologia --- Resistència de materials --- Equacions de Von Kármán --- Elasticity. --- Diffusion --- Solid state physics. --- Mathematical models. --- Physics --- Solids --- Mathematical physics --- Matter --- Statics --- Rheology --- Strains and stresses --- Strength of materials --- Elastic properties --- Young's modulus --- Properties
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